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Free Course Image Calculus

Free online courseCalculus

Duration of the online course: 33 hours and 57 minutes

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Master limits, derivatives, and integrals with a free online calculus course packed with practice—build confidence fast and earn a certificate if available.

In this free course, learn about

  • Core ideas of differential calculus: instantaneous rate of change and tangent slope
  • Compute limits (one-sided, infinite, at infinity) and handle indeterminate forms algebraically
  • Apply Squeeze Theorem and key limit lim(x→0) sin(x)/x, including proof intuition
  • Understand and use the formal epsilon-delta definition of a limit
  • Define derivatives via the limit definition; find slopes and equations of tangent/normal lines
  • Differentiate using power, product, quotient, and chain rules (incl. trig, exp, log forms)
  • Differentiate implicitly and use logarithmic differentiation for x in base and exponent (e.g., x^x)
  • Use l'Hôpital's rule to evaluate limits with 0/0 or ∞/∞ forms
  • Analyze functions with derivatives: extrema, monotonicity, concavity, inflection points, graphing
  • Solve optimization and related rates problems; apply Mean Value Theorem and average value of f
  • Compute indefinite/definite integrals; substitution, parts, trig substitution; area and distance
  • Set up volumes of solids of revolution; basic differential equations and exponential growth models
  • Work with sequences/series: arithmetic sums, Taylor/Maclaurin series, remainder, Euler's formula
  • Multivariable calc: partials, gradient/divergence/curl; multiple, line, Green's, surface integrals

Course Description

Calculus can feel like a new language at first, but once the key ideas click, it becomes one of the most powerful tools you can use to describe change, motion, growth, and optimization. This free online course guides you from the first intuitions behind limits and rates of change to the kinds of problems used in exams and real applications. You will learn to think clearly about how a function behaves near a point, how small input changes create big effects, and how those ideas connect naturally to slopes, areas, and accumulated quantities.

Instead of treating formulas as things to memorize, the course builds understanding step by step: you strengthen your intuition for limits, connect the derivative to tangent lines and instantaneous velocity, and practice using differentiation rules to handle increasingly complex expressions. Along the way, you develop the ability to interpret graphs using derivative information, identify patterns like concavity and inflection points, and solve optimization and related-rates problems with a structured approach that reduces guesswork.

As you progress, integration appears not as a separate topic but as a natural counterpart to differentiation. You will learn to model accumulation with definite integrals, compute antiderivatives with reliable techniques, and apply integrals to geometry and motion. The course also expands your toolbox with series and polynomial approximations, helping you estimate functions and understand why approximations work so well near a point.

For learners ready to go further, the material opens the door to multivariable calculus and vector calculus ideas such as partial derivatives, gradients, divergence, curl, and multiple integrals, building intuition for surfaces and fields in higher dimensions. With plentiful practice opportunities, this course is designed for students aiming to improve school performance, prepare for standardized assessments, or strengthen the math foundation used in physics, engineering, economics, and data-driven fields.

Course content

  • Video class: Newton, Leibniz, and Usain Bolt | Derivatives introduction | AP Calculus AB | Khan Academy 09m
  • Exercise: What fundamental question does differential calculus address?
  • Video class: Introduction to limits | Limits | Differential Calculus | Khan Academy 11m
  • Exercise: Understanding Limits in Calculus
  • Video class: Introduction to limits 2 | Limits | Precalculus | Khan Academy 07m
  • Exercise: What is the limit as x approaches 2 for this modified function?
  • Video class: Limit examples (part 1) | Limits | Differential Calculus | Khan Academy 08m
  • Exercise: What is the limit as x approaches -1 of 2x + 2 over x + 1?
  • Video class: Limit examples (part 2) | Limits | Differential Calculus | Khan Academy 06m
  • Exercise: What is the limit of 1/x as x approaches 0 from the positive side?
  • Video class: Limit examples (part 3) | Limits | Differential Calculus | Khan Academy 09m
  • Exercise: What is the limit of x² as x approaches infinity with a third degree polynomial in the denominator?
  • Video class: Limit examples w/ brain malfunction on first prob (part 4) | Differential Calculus | Khan Academy 15m
  • Exercise: Evaluate the limit \( \lim_{{y \to 5}} \frac{{y^2 - 25}}{{y - 5}} \).
  • Video class: Squeeze theorem (sandwich theorem) | Limits | Differential Calculus | Khan Academy 07m
  • Exercise: What is the Squeeze Theorem?
  • Video class: Proof: lim (sin x)/x | Limits | Differential Calculus | Khan Academy 18m
  • Exercise: What is the limit of sine(x)/x as x approaches 0?
  • Video class: More limits | Limits | Differential Calculus | Khan Academy 13m
  • Exercise: What is the limit?
  • Video class: Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy 12m
  • Exercise: What is the formal definition of a limit in calculus?
  • Video class: Epsilon-delta limit definition 2 | Limits | Differential Calculus | Khan Academy 10m
  • Exercise: What is the limit of the function as x approaches 1?
  • Video class: Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy 15m
  • Exercise: What is the slope of the line that passes through the points (3, 4) and (6, 10)?
  • Video class: Calculating slope of tangent line using derivative definition | Differential Calculus | Khan Academy 08m
  • Exercise: Find the slope of the tangent line at a point on the curve y = x².
  • Video class: The derivative of f(x)=x^2 for any x | Taking derivatives | Differential Calculus | Khan Academy 11m
  • Exercise: What is the derivative of the function y = x² at any point x?
  • Video class: Derivative intuition module | Taking derivatives | Differential Calculus | Khan Academy 03m
  • Exercise: What is the slope of the tangent line for f(x) = 2x^3 at x = 0?
  • Video class: Calculus: Derivatives 1 | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Exercise: What is the derivative of a function at a point?
  • Video class: Calculus: Derivatives 2 | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Exercise: What is the slope of the function f(x) = x² at x = 3?
  • Video class: Power rule introduction (old) | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Exercise: If the function f(x) = 7x^3 - 2x^2 + 5x + 9, what is the derivative f'(x)?
  • Video class: The Chain Rule 09m
  • Exercise: What is the derivative of f(x) = 2x + 3 raised to the fifth power using the chain rule?
  • Video class: Chain Rule Examples 09m
  • Video class: Even More Chain Rule 09m
  • Video class: Product rule | Taking derivatives | Differential Calculus | Khan Academy 08m
  • Video class: Quotient rule and common derivatives | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Video class: Derivatives (part 9) 09m
  • Exercise: If h(x) = cos(2x) * (3x^2 + 4x)^5, which rule should we apply first to find h'(x)?
  • Video class: Proof: d/dx(x^n) | Taking derivatives | Differential Calculus | Khan Academy 07m
  • Video class: Proof: d/dx(sqrt(x)) | Taking derivatives | Differential Calculus | Khan Academy 05m
  • Video class: Proof: d/dx(ln x) = 1/x | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Video class: Proof: d/dx(e^x) = e^x | Taking derivatives | Differential Calculus | Khan Academy 04m
  • Video class: Proofs of derivatives of ln(x) and e^x | Taking derivatives | Differential Calculus | Khan Academy 12m
  • Video class: Extreme derivative word problem (advanced) | Differential Calculus | Khan Academy 23m
  • Exercise: For a given parabola y = x^2, consider a normal line at any point on the curve. What property does the normal line have with respect to the curve?
  • Video class: Implicit Differentiation 15m
  • Video class: Implicit Differentiation (part 2) 10m
  • Video class: More implicit differentiation 11m
  • Video class: More chain rule and implicit differentiation intuition 10m
  • Video class: Trig Implicit Differentiation Example 12m
  • Video class: Derivative of x^(x^x) | Taking derivatives | Differential Calculus | Khan Academy 09m
  • Exercise: Which of the following methods is commonly used to differentiate the function y = x^x, where x is both the base and the exponent?
  • Video class: Introduction to l'Hôpital's rule | Derivative applications | Differential Calculus | Khan Academy 08m
  • Video class: L'Hôpital's rule example 1 | Derivative applications | Differential Calculus | Khan Academy 07m
  • Video class: L'Hôpital's rule example 2 | Derivative applications | Differential Calculus | Khan Academy 05m
  • Video class: L'Hôpital's rule example 3 | Derivative applications | Differential Calculus | Khan Academy 07m
  • Video class: Maxima Minima Slope Intuition 09m
  • Video class: Inflection Points and Concavity Intuition 13m
  • Exercise: What is the primary condition for a point on a curve to be classified as an inflection point?
  • Video class: Monotonicity Theorem 09m
  • Video class: Calculus: Maximum and minimum values on an interval 11m
  • Video class: Graphing using derivatives | Derivative applications | Differential Calculus | Khan Academy 20m
  • Video class: Another example graphing with derivatives | Differential Calculus | Khan Academy 25m
  • Video class: Graphing with Calculus 09m
  • Video class: Optimization with Calculus 1 09m
  • Exercise: Given a function f(x) that represents the travel cost of a substance being transported, under what condition would the travel cost be minimized?
  • Video class: Optimization with Calculus 2 11m
  • Video class: Optimization with Calculus 3 17m
  • Video class: Optimization Example 4 09m
  • Video class: Introduction to rate-of-change problems 09m
  • Video class: Equation of a tangent line | Taking derivatives | Differential Calculus | Khan Academy 08m
  • Video class: Rates-of-change (part 2) 05m
  • Exercise: If the radius of a ripple in a pond is expanding at a constant rate of 4 meters per second, how fast is the area of the ripple increasing when the radius is 5 meters?
  • Video class: Ladder rate-of-change problem 10m
  • Video class: Mean value theorem | Derivative applications | Differential Calculus | Khan Academy 16m
  • Video class: The Indefinite Integral or Anti-derivative 09m
  • Video class: Indefinite integrals (part II) 09m
  • Video class: Indefinite Integration (part III) 09m
  • Video class: Indefinite Integration (part IV) 09m
  • Exercise: What is the result of using the integration by substitution method on the integral ∫(3x+1)(x^2+2x+5)^4 dx?
  • Video class: Indefinite Integration (part V) 09m
  • Video class: Integration by Parts (part 6 of Indefinite Integration) 09m
  • Video class: Indefinite Integration (part 7) 09m
  • Video class: Another u-subsitution example 05m
  • Video class: Introduction to definite integrals 09m
  • Video class: Definite integrals (part II) 09m
  • Exercise: If you are given a velocity function v(t) = 32t and you want to find the distance traveled by an object from time t = 0 to t = 5 seconds, which of the following expressions correctly represents the distance after 5 seconds?
  • Video class: Definite Integrals (area under a curve) (part III) 09m
  • Video class: Definite Integrals (part 4) 09m
  • Video class: Definite Integrals (part 5) 09m
  • Video class: Definite integral with substitution 08m
  • Video class: Integrals: Trig Substitution 1 07m
  • Video class: Integrals: Trig Substitution 2 08m
  • Exercise: Evaluate the indefinite integral of 1/(25 + x^2) dx.
  • Video class: Integrals: Trig Substitution 3 (long problem) 17m
  • Video class: Periodic Definite Integral 15m
  • Video class: Simple Differential Equations 14m
  • Video class: Solid of Revolution (part 1) 10m
  • Video class: Solid of Revolution (part 2) 07m
  • Video class: Solid of Revolution (part 3) 08m
  • Exercise: What is the formula for the volume of a sphere with radius r?
  • Video class: Solid of Revolution (part 4) 08m
  • Video class: Solid of Revolution (part 5) 09m
  • Video class: Solid of Revolution (part 6) 09m
  • Video class: Solid of Revolution (part 7) 10m
  • Video class: Solid of Revolution (part 8) 04m
  • Video class: Sequences and Series (part 1) 09m
  • Exercise: What is the sum of the arithmetic series described by the sequence where the first term is 1 and the last term is 50?
  • Video class: Sequences and series (part 2) 10m
  • Video class: Taylor 12m
  • Video class: Maclaurin series of cos(x) | Series | AP Calculus BC | Khan Academy 05m
  • Video class: Maclaurin series of sin(x) | Series | AP Calculus BC | Khan Academy 06m
  • Video class: Maclaurin series of e_ | Series | AP Calculus BC | Khan Academy 06m
  • Video class: Euler's formula 11m
  • Exercise: In the context of Euler's formula, how can we define the relationship between the exponential function and trigonometric functions when using imaginary exponents?
  • Video class: Visualizing Taylor series approximations | Series | AP Calculus BC | Khan Academy 06m
  • Video class: Taylor 07m
  • Video class: Visualizing Taylor polynomial approximations | AP Calculus BC | Khan Academy 06m
  • Video class: Polynomial approximation of functions (part 1) 09m
  • Video class: Polynomial approximation of functions (part 2) 10m
  • Video class: Approximating functions with polynomials (part 3) 07m
  • Exercise: What is the Maclaurin series expansion of the function e^x?
  • Video class: Polynomial approximation of functions (part 4) 10m
  • Video class: Polynomial approximations of functions (part 5) 09m
  • Video class: Polynomial approximation of functions (part 6) 09m
  • Video class: Polynomial approximation of functions (part 7) 10m
  • Video class: Taylor Polynomials 18m
  • Video class: Exponential Growth 16m
  • Exercise: Consider a population of fungi that grows exponentially and initially contains 200 fungi. After 2 hours, the population grows to 800 fungi. What will be the population size after 5 hours, assuming the growth rate remains constant?
  • Video class: AP Calculus BC exams: 2008 1 a | AP Calculus BC | Khan Academy 08m
  • Video class: AP Calculus BC exams: 2008 1 b 09m
  • Video class: AP Calculus BC exams: 2008 1 c 09m
  • Video class: AP Calculus BC exams: 2008 1 d | AP Calculus BC | Khan Academy 05m
  • Video class: Calculus BC 2008 2 a | AP Calculus BC | Khan Academy 08m
  • Video class: Calculus BC 2008 2 b 10m
  • Exercise: What is the average value of a function f(x) over the interval [a, b] in terms of area under the curve?
  • Video class: Calculus BC 2008 2d | AP Calculus BC | Khan Academy 04m
  • Video class: Partial derivatives | Multivariable Calculus | Khan Academy 11m
  • Video class: Partial derivatives 2 | Multivariable Calculus | Khan Academy 10m
  • Video class: Gradient 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy 09m
  • Video class: Gradient of a scalar field | Multivariable Calculus | Khan Academy 10m
  • Video class: Divergence 1 | Multivariable Calculus | Khan Academy 10m
  • Exercise: What does the divergence of a vector field represent?
  • Video class: Divergence 2 | Multivariable Calculus | Khan Academy 10m
  • Video class: Divergence 3 | Multivariable Calculus | Khan Academy 10m
  • Video class: Curl 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy 09m
  • Video class: Curl 2 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy 10m
  • Video class: Curl 3 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus | Khan Academy 10m
  • Video class: Double integral 1 | Double and triple integrals | Multivariable Calculus | Khan Academy 10m
  • Exercise: What is the primary intuition behind using definite integrals to find the area under a curve?
  • Video class: Double integrals 2 | Double and triple integrals | Multivariable Calculus | Khan Academy 09m
  • Video class: Double integrals 3 | Double and triple integrals | Multivariable Calculus | Khan Academy 08m
  • Video class: Double integrals 4 | Double and triple integrals | Multivariable Calculus | Khan Academy 09m
  • Video class: Double integrals 5 | Double and triple integrals | Multivariable Calculus | Khan Academy 09m
  • Video class: Double integrals 6 | Double and triple integrals | Multivariable Calculus | Khan Academy 09m
  • Video class: Triple integrals 1 | Double and triple integrals | Multivariable Calculus | Khan Academy 10m
  • Exercise: What is the result of calculating the triple integral with the respective limits for the coordinates of the rectangular prism given by the boundaries 0 ≤ x ≤ 3, 0 ≤ y ≤ 4, and 0 ≤ z ≤ 2?
  • Video class: Triple integrals 2 | Double and triple integrals | Multivariable Calculus | Khan Academy 07m
  • Video class: Triple integrals 3 | Double and triple integrals | Multivariable Calculus | Khan Academy 11m
  • Video class: (2^ln x)/x Antiderivative Example 08m
  • Video class: Introduction to the line integral | Multivariable Calculus | Khan Academy 18m
  • Video class: Line integral example 1 | Line integrals and Green's theorem | Multivariable Calculus | Khan Academy 13m
  • Video class: Line integral example 2 (part 1) | Multivariable Calculus | Khan Academy 12m
  • Exercise: Consider the function f(x, y) = x + y². Imagine you are asked to compute the surface area of a 3D shape formed by traveling from the point (2, 0), making a full circle of radius 2 in the xy-plane. What type of integral is necessary to determine the surface area of the walls of this 3D shape?
  • Video class: Line integral example 2 (part 2) | Multivariable Calculus | Khan Academy 09m
  • Video class: Position vector valued functions | Multivariable Calculus | Khan Academy 07m
  • Video class: Derivative of a position vector valued function | Multivariable Calculus | Khan Academy 14m
  • Video class: Differential of a vector valued function | Multivariable Calculus | Khan Academy 05m
  • Video class: Vector valued function derivative example | Multivariable Calculus | Khan Academy 12m
  • Video class: Line integrals and vector fields | Multivariable Calculus | Khan Academy 16m
  • Exercise: A particle is pulled across a frictionless surface with a force of 8 newtons at an angle of 45 degrees to the direction of displacement. If the particle is displaced 10 meters, calculate the work done on the particle.
  • Video class: Using a line integral to find the work done by a vector field example | Khan Academy 11m
  • Video class: Parametrization of a reverse path | Khan Academy 07m
  • Video class: Scalar field line integral independent of path direction | Multivariable Calculus | Khan Academy 16m
  • Video class: Vector field line integrals dependent on path direction | Multivariable Calculus | Khan Academy 15m
  • Video class: Path independence for line integrals | Multivariable Calculus | Khan Academy 17m
  • Video class: Closed curve line integrals of conservative vector fields | Multivariable Calculus | Khan Academy 08m
  • Exercise: If a vector field \( F \) is conservative, which of the following statements is true concerning the line integral of \( F \) over a closed path \( C \)?
  • Video class: Example of closed line integral of conservative field | Multivariable Calculus | Khan Academy 11m
  • Video class: Second example of line integral of conservative vector field | Multivariable Calculus | Khan Academy 10m
  • Video class: Green's theorem proof part 1 | Multivariable Calculus | Khan Academy 14m
  • Video class: Green's theorem proof (part 2) | Multivariable Calculus | Khan Academy 19m
  • Video class: Green's theorem example 1 | Multivariable Calculus | Khan Academy 10m
  • Video class: Green's theorem example 2 | Multivariable Calculus | Khan Academy 07m
  • Exercise: Consider a closed path on the xy-plane, described by the unit circle equation x^2 + y^2 = 1, traversed in the clockwise direction. Using Green's theorem, which of the following represents the result of the line integral of the vector field f(x, y) = (2y, -3x) around this path?
  • Video class: Introduction to parametrizing a surface with two parameters | Multivariable Calculus | Khan Academy 19m
  • Video class: Determining a position vector-valued function for a parametrization of two parameters | Khan Academy 16m
  • Video class: Partial derivatives of vector-valued functions | Multivariable Calculus | Khan Academy 10m
  • Video class: Introduction to the surface integral | Multivariable Calculus | Khan Academy 22m
  • Video class: Example of calculating a surface integral part 1 | Multivariable Calculus | Khan Academy 10m
  • Video class: Example of calculating a surface integral part 2 | Multivariable Calculus | Khan Academy 10m
  • Exercise: When calculating the surface area of a torus using a surface integral, we need to determine the cross product of two partial derivatives of the parameterization. Once the cross product is calculated, what is the subsequent step required to find the surface area?
  • Video class: Example of calculating a surface integral part 3 | Multivariable Calculus | Khan Academy 09m
  • Video class: 2011 Calculus AB free response #1a | AP Calculus AB solved exams | AP Calculus AB | Khan Academy 07m
  • Video class: 2011 Calculus AB Free Response #1 parts b c d | AP Calculus AB | Khan Academy 14m
  • Video class: 2011 Calculus AB free response #2 (a 09m
  • Video class: 2011 Calculus AB free response #2 (c 07m
  • Video class: 2011 Calculus AB free response #3 (a 08m
  • Exercise: Consider the region R bounded by the function h(x) = x^2 and the horizontal line y = 4 between x = 0 and some positive x-value where these plots intersect. Which of the following integrals gives the area of region R?
  • Video class: 2011 Calculus AB free response #3 (c) | AP Calculus AB | Khan Academy 06m
  • Video class: 2011 Calculus AB free response #4a | AP Calculus AB | Khan Academy 04m
  • Video class: 2011 Calculus AB free response #4b | AP Calculus AB | Khan Academy 13m
  • Video class: 2011 Calculus AB free response #4c | AP Calculus AB | Khan Academy 04m
  • Video class: 2011 Calculus AB free response #4d | AP Calculus AB | Khan Academy 05m
  • Video class: 2011 Calculus AB free response #5a | AP Calculus AB | Khan Academy 06m
  • Exercise: Given a continuously growing function G(t) that models the energy consumption in a city as a function of time t (years since 2020), which of the following differential equations correctly describes a situation where the rate of growth of energy consumption is proportional to the difference between the current consumption and a baseline consumption of 500 units?
  • Video class: 2011 Calculus AB free response #5b | AP Calculus AB | Khan Academy 08m
  • Video class: 2011 Calculus AB free response #5c. | AP Calculus AB | Khan Academy 09m
  • Video class: 2011 Calculus AB free response #6a | AP Calculus AB | Khan Academy 04m
  • Video class: 2011 Calculus AB free response #6b | AP Calculus AB | Khan Academy 05m
  • Video class: 2011 Calculus AB free response #6c | AP Calculus AB | Khan Academy 07m
  • Video class: 2011 Calculus BC free response #1a | AP Calculus BC solved exams | AP Calculus BC | Khan Academy 06m
  • Exercise: At time t, a particle moving in the xy-plane is described with position derivatives where dx/dt = 4t + 1 and dy/dt = sin(t^2). At t=0, x(0)=0 and y(0)=-4. What is the y-component of the particle's acceleration vector at time t=2?
  • Video class: 2011 Calculus BC free response #1 (b 07m
  • Video class: 2011 Calculus BC free response #1d | AP Calculus BC | Khan Academy 08m
  • Video class: 2011 Calculus BC free response #3a | AP Calculus BC | Khan Academy 05m
  • Video class: 2011 Calculus BC free response #3 (b 08m
  • Video class: 2011 Calculus BC free response #6a | AP Calculus BC | Khan Academy 07m
  • Video class: 2011 Calculus BC free response #6b | AP Calculus BC | Khan Academy 09m
  • Exercise: What are the first four non-zero terms of the Taylor series for cosine of x about x equals 0?
  • Video class: 2011 Calculus BC free response #6c | AP Calculus BC | Khan Academy 02m
  • Video class: Taylor polynomial remainder (part 1) | Series | AP Calculus BC | Khan Academy 11m
  • Video class: Taylor polynomial remainder (part 2) | Series | AP Calculus BC | Khan Academy 15m
  • Video class: 2011 Calculus BC free response #6d | AP Calculus BC | Khan Academy 11m
  • Video class: Constructing a unit normal vector to a curve | Multivariable Calculus | Khan Academy 10m
  • Video class: 2D divergence theorem | Line integrals and Green's theorem | Multivariable Calculus | Khan Academy 14m
  • Exercise: In a two-dimensional vector field, what does the divergence measure at a point?
  • Video class: Conceptual clarification for 2D divergence theorem | Multivariable Calculus | Khan Academy 09m
  • Video class: Surface integral example part 2: Calculating the surface differential | Khan Academy 12m
  • Video class: Surface integral example part 1: Parameterizing the unit sphere | Khan Academy 11m
  • Video class: Surface integral example part 3: The home stretch | Multivariable Calculus | Khan Academy 12m
  • Video class: Surface integral ex2 part 1: Parameterizing the surface | Multivariable Calculus | Khan Academy 05m
  • Video class: Surface integral ex2 part 2: Evaluating integral | Multivariable Calculus | Khan Academy 09m
  • Exercise: Given a surface integral in the form \( \iint_S f(x, y, z) \, dS \), if the parameterized surface \( r(u, v) \) is defined, how do we express \( dS \) in terms of \( du \) and \( dv \)?
  • Video class: Surface integral ex3 part 1: Parameterizing the outside surface | Khan Academy 09m
  • Video class: Surface integral ex3 part 2: Evaluating the outside surface | Multivariable Calculus | Khan Academy 09m
  • Video class: Surface integral ex3 part 3: Top surface | Multivariable Calculus | Khan Academy 12m
  • Video class: Surface integral ex3 part 4: Home stretch | Multivariable Calculus | Khan Academy 04m
  • Video class: Conceptual understanding of flux in three dimensions | Multivariable Calculus | Khan Academy 08m
  • Video class: Constructing a unit normal vector to a surface | Multivariable Calculus | Khan Academy 06m
  • Exercise: What does the cross product of the partial derivatives of the position vector function with respect to u and v represent on a surface?

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