Integral Calculus
Learn integral calculus—indefinite integrals, Riemann sums, definite integrals, application problems, and more.
5 units232 videos25 hours
Learn it live with a class on Puzzicle · Teachers: use it with your class
Unit 1: Integrals
Accumulations of change introduction
Approximation with Riemann sums
Introducing Riemann approximations ✓ Check-ins ✎ Fixed a slip6:45
Over- and under-estimating of Riemann sums ✓ Check-ins4:01
Working an example: finding a Riemann sum using a table ✓ Check-ins6:39
Working an example: over- and under-estimation of Riemann sums ✓ Check-ins4:42
Approximating area using midpoint sums ✓ Check-ins5:29
Approximating area using trapezoidal sums ✓ Check-ins8:27
Motion problem with Riemann sum approximation ✓ Check-ins10:00
Summation notation review
Riemann sums in summation notation
Defining integrals with Riemann sums
Fundamental theorem of calculus and accumulation functions
Introducing the fundamental theorem of calculus and accumulation functions ✓ Check-ins ✎ Fixed a slip8:03
Understanding functions defined by definite integrals (accumulation functions) ✓ Check-ins4:17
Finding derivative with fundamental theorem of calculus ✓ Check-ins3:24
Finding derivative with fundamental theorem of calculus: chain rule ✓ Check-ins3:07
Interpreting the behavior of accumulation functions
Properties of definite integrals
Understanding definite integrals with a negative value ✓ Check-ins5:24
Finding definite integrals using area formulas ✓ Check-ins4:17
Evaluating a definite integral over a single point ✓ Check-ins2:31
Integrating scaled version of function ✓ Check-ins5:21
Switching bounds of definite integral ✓ Check-ins4:37
Integrating sums of functions ✓ Check-ins6:46
Working examples: Finding definite integrals using algebraic properties ✓ Check-ins2:19
Evaluating definite integrals on adjacent intervals ✓ Check-ins3:06
Working an example: Breaking up the integral's interval ✓ Check-ins ✎ Fixed a slip7:25
Working an example: Merging definite integrals over adjacent intervals ✓ Check-ins4:02
Evaluating functions defined by integrals: switched interval ✓ Check-ins2:54
Finding derivative with fundamental theorem of calculus: x is on lower bound ✓ Check-ins3:00
Finding derivative with fundamental theorem of calculus: x is on both bounds ✓ Check-ins5:04
Functions defined by integrals: challenge problem ✓ Check-ins9:45
Fundamental theorem of calculus and definite integrals
Reverse power rule
Indefinite integrals of common functions
Definite integrals of common functions
Evaluating definite integrals: reverse power rule ✓ Check-ins4:14
Evaluating definite integral of a rational function ✓ Check-ins5:05
Evaluating definite integral of a radical function ✓ Check-ins3:51
Evaluating definite integral of a trig function ✓ Check-ins5:00
Evaluating definite integral involving a natural log function ✓ Check-ins7:27
Evaluating definite integral of a piecewise function ✓ Check-ins6:14
Evaluating definite integral of an absolute value function ✓ Check-ins6:53
Integrating with u-substitution
Introducing 𝘶-substitution ✓ Check-ins5:11
Using 𝘶-substitution: multiplying by a constant ✓ Check-ins5:33
Using 𝘶-substitution: defining 𝘶 ✓ Check-ins3:35
Using 𝘶-substitution: defining 𝘶 (more examples) ✓ Check-ins4:34
Using 𝘶-substitution: rational function ✓ Check-ins4:04
Using 𝘶-substitution: logarithmic function ✓ Check-ins3:39
Using 𝘶-substitution: definite integrals ✓ Check-ins5:53
Using 𝘶-substitution: definite integral of exponential function ✓ Check-ins ✎ Fixed a slip7:53
𝘶-substitution: special application ✓ Check-ins5:01
𝘶-substitution: double substitution ✓ Check-ins8:57
𝘶-substitution: challenging application ✓ Check-ins8:40
Integrating using long division and completing the square
Integrating using trigonometric identities
Trigonometric substitution
Introduction to trigonometric substitution ✓ Check-ins8:52
Substitution with x=sin(theta) ✓ Check-ins5:08
More trig sub practice ✓ Check-ins7:38
Trig and u substitution together (part 1) ✓ Check-ins8:55
Trig and u substitution together (part 2) ✓ Check-ins5:45
Trig substitution with tangent ✓ Check-ins4:46
More trig substitution with tangent ✓ Check-ins8:11
Long trig sub problem ✓ Check-ins17:52
Integration by parts
Integration by parts intro ✓ Check-ins3:52
Integration by parts: ∫x⋅cos(x)dx ✓ Check-ins3:52
Integration by parts: ∫ln(x)dx ✓ Check-ins ✎ Fixed a slip3:50
Integration by parts: ∫x²⋅𝑒ˣdx ✓ Check-ins ✎ Fixed a slip6:46
Integration by parts: ∫𝑒ˣ⋅cos(x)dx ✓ Check-ins ✎ Fixed a slip7:17
Integration by parts: definite integrals ✓ Check-ins5:43
Integrating using linear partial fractions
Improper integrals
Unit 2: Differential equations
Differential equations introduction
Verifying solutions for differential equations
Sketching slope fields
Reasoning using slope fields
Approximation with Euler’s method
Separation of variables
Particular solutions to differential equations
Finding particular solutions to differential equations: rational function ✓ Check-ins ✎ Fixed a slip4:24
Finding particular solutions to differential equations: exponential function ✓ Check-ins3:08
Working an example: finding a specific solution to a separable equation ✓ Check-ins5:38
Working an example: separable equation with an implicit solution ✓ Check-ins3:18
Exponential models
Logistic models
Growth models: introduction ✓ Check-ins9:37
The logistic growth model ✓ Check-ins ✎ Fixed a slip10:49
Worked example: Logistic model word problem ✓ Check-ins8:47
Logistic equations (Part 1) ✓ Check-ins13:38
Logistic equations (Part 2) ✓ Check-ins9:58
Worked example: logistic model equations ✓ Check-ins ✎ Fixed a slip9:34
Unit 3: Applications of integrals
Average value of a function
Straight-line motion
Solving motion problems with integrals: displacement vs. distance ✓ Check-ins8:04
Analyzing motion problems: position ✓ Check-ins2:56
Analyzing motion problems: total distance traveled ✓ Check-ins2:53
Working an example: motion problems (with definite integrals) ✓ Check-ins ✎ Fixed a slip8:32
Finding the average acceleration over interval ✓ Check-ins5:07
Non-motion applications of integrals
Determining area under rate function gives the net change ✓ Check-ins7:27
Interpreting definite integral as net change ✓ Check-ins5:16
Working examples: interpreting definite integrals in context ✓ Check-ins4:21
Analyzing problems involving definite integrals ✓ Check-ins5:52
Working an example: problem involving definite integral (algebraic) ✓ Check-ins7:50
Area: vertical area between curves
Calculating area between a curve and the x-axis ✓ Check-ins5:54
Calculating area between a curve and the x-axis: negative area ✓ Check-ins7:28
Calculating the area between curves ✓ Check-ins6:50
Working an example: area between curves ✓ Check-ins ✎ Fixed a slip5:25
Calculating the composite area between curves ✓ Check-ins6:54
Area: horizontal area between curves
Volume: squares and rectangles cross sections
Volume: triangles and semicircles cross sections
Volume: disc method (revolving around x- and y-axes)
Volume: disc method (revolving around other axes)
Volume: washer method (revolving around x- and y-axes)
Volume: washer method (revolving around other axes)
Using the washer method: rotating around horizontal line (not x-axis), part 1 ✓ Check-ins6:38
Using the washer method: rotating around horizontal line (not x-axis), part 2 ✓ Check-ins8:14
Using the washer method: rotating around vertical line (not y-axis), part 1 ✓ Check-ins7:50
Using the washer method: rotating around vertical line (not y-axis), part 2 ✓ Check-ins5:35
Unit 4: Parametric equations, polar coordinates, and vector-valued functions
Arc length: parametric curves
Planar motion
Area: polar regions (single curve)
Area: polar regions (two curves)
Arc length: polar curves
Calculator-active practice
Unit 5: Series
Convergent and divergent infinite series
Convergent and divergent sequences ✓ Check-ins4:59
Worked example: sequence convergence/divergence ✓ Check-ins3:53
Partial sums intro ✓ Check-ins6:09
Partial sums: formula for nth term from partial sum ✓ Check-ins4:05
Partial sums: term value from partial sum ✓ Check-ins4:09
Infinite series as limit of partial sums ✓ Check-ins4:48
Infinite geometric series
Worked example: convergent geometric series ✓ Check-ins3:50
Worked example: divergent geometric series ✓ Check-ins3:14
Infinite geometric series word problem: bouncing ball ✓ Check-ins5:10
Infinite geometric series word problem: repeating decimal ✓ Check-ins8:02
Convergent & divergent geometric series (with manipulation) ✓ Check-ins7:08
nth-term test
Integral test
Harmonic series and p-series
Comparison tests
Alternating series test
Absolute and conditional convergence
Alternating series error bound
Taylor and Maclaurin polynomials intro
Taylor & Maclaurin polynomials intro (part 1) ✓ Check-ins12:59
Taylor & Maclaurin polynomials intro (part 2) ✓ Check-ins7:27
Worked example: Maclaurin polynomial ✓ Check-ins4:43
Worked example: coefficient in Maclaurin polynomial ✓ Check-ins3:05
Worked example: coefficient in Taylor polynomial ✓ Check-ins3:53
Visualizing Taylor polynomial approximations ✓ Check-ins6:38
Lagrange error bound
Power series intro
Function as a geometric series
Maclaurin series of eˣ, sin(x), and cos(x)
Maclaurin series of cos(x) ✓ Check-ins5:37
Maclaurin series of sin(x) ✓ Check-ins6:33
Maclaurin series of eˣ ✓ Check-ins6:10
Worked example: power series from cos(x) ✓ Check-ins8:01
Worked example: cosine function from power series ✓ Check-ins ✎ Fixed a slip6:09
Worked example: recognizing function from Taylor series ✓ Check-ins4:26
Visualizing Taylor series approximations ✓ Check-ins6:01
Euler's formula & Euler's identity ✓ Check-ins11:27
Representing functions as power series
Integrating power series ✓ Check-ins6:05
Differentiating power series ✓ Check-ins ✎ Fixed a slip8:02
Finding function from power series by integrating ✓ Check-ins5:47
Interval of convergence for derivative and integral ✓ Check-ins ✎ Fixed a slip8:35
Converting explicit series terms to summation notation ✓ Check-ins3:16
Converting explicit series terms to summation notation (n ≥ 2) ✓ Check-ins4:25
Proof videos
Formal definition for limit of a sequence ✓ Check-ins4:49
Proving a sequence converges using the formal definition ✓ Check-ins5:31
Finite geometric series formula ✓ Check-ins7:15
Infinite geometric series formula intuition ✓ Check-ins7:54
Proof of infinite geometric series as a limit ✓ Check-ins4:46
Proof of p-series convergence criteria ✓ Check-ins ✎ Fixed a slip9:37
Learn this with a class
With a class on Puzzicle this course runs as a timetable of live study parties: your batch watches together and answers the check-ins, sometimes with a teacher leading.
Course content © Khan Academy, used under CC BY-NC-SA. See this course on Khan Academy.