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Unit: Integrals

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Possible mastery points

Accumulations of change introduction

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Introduction to integral calculus
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Definite integrals intro
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Exploring accumulation of change
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Worked example: accumulation of change
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Practice
Accumulation of changeGet 3 of 4 questions to level up!

Approximation with Riemann sums

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Riemann approximation introduction
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Over- and under-estimation of Riemann sums
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Left & right Riemann sums
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Worked example: finding a Riemann sum using a table
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Worked example: over- and under-estimation of Riemann sums
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Midpoint sums
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Trapezoidal sums
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Understanding the trapezoidal rule
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Riemann sums review
(Opens a modal)
Motion problem with Riemann sum approximation
(Opens a modal)
Practice
Left & right Riemann sumsGet 3 of 4 questions to level up!
Over- and under-estimation of Riemann sumsGet 3 of 4 questions to level up!
Midpoint & trapezoidal sumsGet 3 of 4 questions to level up!

Summation notation review

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Summation notation
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Summation notation
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Worked examples: Summation notation
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Practice
Summation notationGet 3 of 4 questions to level up!

Riemann sums in summation notation

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Riemann sums in summation notation
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Riemann sums in summation notation
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Worked example: Riemann sums in summation notation
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Midpoint and trapezoidal sums in summation notation
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Riemann sums in summation notation: challenge problem
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Practice
Riemann sums in summation notationGet 3 of 4 questions to level up!

Defining integrals with Riemann sums

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Definite integral as the limit of a Riemann sum
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Definite integral as the limit of a Riemann sum
(Opens a modal)
Worked example: Rewriting definite integral as limit of Riemann sum
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Worked example: Rewriting limit of Riemann sum as definite integral
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Practice
Definite integral as the limit of a Riemann sumGet 3 of 4 questions to level up!

Quiz 1

Level up on the above skills and collect up to 700 Mastery points

Fundamental theorem of calculus and accumulation functions

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The fundamental theorem of calculus and accumulation functions
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Functions defined by definite integrals (accumulation functions)
(Opens a modal)
Finding derivative with fundamental theorem of calculus
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Finding derivative with fundamental theorem of calculus: chain rule
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Practice
Functions defined by definite integrals (accumulation functions)Get 3 of 4 questions to level up!
Finding derivative with fundamental theorem of calculusGet 3 of 4 questions to level up!
Finding derivative with fundamental theorem of calculus: chain ruleGet 3 of 4 questions to level up!

Interpreting the behavior of accumulation functions

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Interpreting the behavior of accumulation functions
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Interpreting the behavior of accumulation functions
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Practice
Interpreting the behavior of accumulation functionsGet 3 of 4 questions to level up!

Properties of definite integrals

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Negative definite integrals
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Finding definite integrals using area formulas
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Definite integral over a single point
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Integrating scaled version of function
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Switching bounds of definite integral
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Integrating sums of functions
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Worked examples: Finding definite integrals using algebraic properties
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Definite integrals on adjacent intervals
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Worked example: Breaking up the integral's interval
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Worked example: Merging definite integrals over adjacent intervals
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Functions defined by integrals: switched interval
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Finding derivative with fundamental theorem of calculus: x is on lower bound
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Finding derivative with fundamental theorem of calculus: x is on both bounds
(Opens a modal)
Functions defined by integrals: challenge problem
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Definite integrals properties review
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Practice
Finding definite integrals using area formulasGet 3 of 4 questions to level up!
Finding definite integrals using algebraic propertiesGet 3 of 4 questions to level up!
Definite integrals over adjacent intervalsGet 3 of 4 questions to level up!

Quiz 2

Level up on the above skills and collect up to 700 Mastery points

Fundamental theorem of calculus and definite integrals

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The fundamental theorem of calculus and definite integrals
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Antiderivatives and indefinite integrals
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Proof of fundamental theorem of calculus
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Practice
The fundamental theorem of calculus and definite integralsGet 3 of 4 questions to level up!
Antiderivatives and indefinite integralsGet 3 of 4 questions to level up!

Reverse power rule

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Reverse power rule
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Indefinite integrals: sums & multiples
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Rewriting before integrating
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Rewriting before integrating: challenge problem
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Reverse power rule review
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Practice
Reverse power ruleGet 3 of 4 questions to level up!
Reverse power rule: negative and fractional powersGet 3 of 4 questions to level up!
Reverse power rule: sums & multiplesGet 3 of 4 questions to level up!
Reverse power rule: rewriting before integratingGet 3 of 4 questions to level up!

Quiz 3

Level up on the above skills and collect up to 600 Mastery points

Indefinite integrals of common functions

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Indefinite integral of 1/x
(Opens a modal)
Indefinite integrals of sin(x), cos(x), and eˣ
(Opens a modal)
Common integrals review
(Opens a modal)
Practice
Indefinite integrals: eˣ & 1/xGet 3 of 4 questions to level up!
Indefinite integrals: sin & cosGet 3 of 4 questions to level up!
Integrating trig functionsGet 5 of 7 questions to level up!

Definite integrals of common functions

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Definite integrals: reverse power rule
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Definite integral of rational function
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Definite integral of radical function
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Definite integral of trig function
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Definite integral involving natural log
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Definite integral of piecewise function
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Definite integral of absolute value function
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Practice
Definite integrals: reverse power ruleGet 3 of 4 questions to level up!
Definite integrals: common functionsGet 3 of 4 questions to level up!
Definite integrals of piecewise functionsGet 3 of 4 questions to level up!

Quiz 4

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Integrating with u-substitution

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𝘶-substitution intro
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𝘶-substitution: multiplying by a constant
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𝘶-substitution: defining 𝘶
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𝘶-substitution: defining 𝘶 (more examples)
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𝘶-substitution
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𝘶-substitution: rational function
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𝘶-substitution: logarithmic function
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𝘶-substitution warmup
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𝘶-substitution: definite integrals
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𝘶-substitution with definite integrals
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𝘶-substitution: definite integral of exponential function
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𝘶-substitution: special application
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𝘶-substitution: double substitution
(Opens a modal)
𝘶-substitution: challenging application
(Opens a modal)
Practice
𝘶-substitution: defining 𝘶Get 3 of 4 questions to level up!
𝘶-substitution: indefinite integralsGet 3 of 4 questions to level up!
𝘶-substitution: definite integralsGet 3 of 4 questions to level up!

Integrating using long division and completing the square

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Integration using long division
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Integration using completing the square and the derivative of arctan(x)
(Opens a modal)
Practice
Integration using long divisionGet 3 of 4 questions to level up!
Integration using completing the squareGet 3 of 4 questions to level up!

Integrating using trigonometric identities

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Integral of cos^3(x)
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Integral of sin^2(x) cos^3(x)
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Integral of sin^4(x)
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Practice
Integration using trigonometric identitiesGet 3 of 4 questions to level up!

Quiz 5

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Trigonometric substitution

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Introduction to trigonometric substitution
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Substitution with x=sin(theta)
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More trig sub practice
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Trig and u substitution together (part 1)
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Trig and u substitution together (part 2)
(Opens a modal)
Trig substitution with tangent
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More trig substitution with tangent
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Long trig sub problem
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Practice
Trigonometric substitutionGet 3 of 4 questions to level up!

Integration by parts

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Integration by parts intro
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Integration by parts: ∫x⋅cos(x)dx
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Integration by parts: ∫ln(x)dx
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Integration by parts: ∫x²⋅𝑒ˣdx
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Integration by parts: ∫𝑒ˣ⋅cos(x)dx
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Integration by parts: definite integrals
(Opens a modal)
Integration by parts challenge
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Integration by parts review
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Practice
Integration by partsGet 3 of 4 questions to level up!
Integration by parts: definite integralsGet 3 of 4 questions to level up!

Integrating using linear partial fractions

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Integration with partial fractions
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Practice
Integration with partial fractionsGet 3 of 4 questions to level up!

Improper integrals

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Introduction to improper integrals
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Divergent improper integral
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Improper integrals review
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Improper integral with two infinite bounds
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Practice
Improper integralsGet 3 of 4 questions to level up!

Quiz 6

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Proof videos

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Proof of fundamental theorem of calculus
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Intuition for second part of fundamental theorem of calculus
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Unit test

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About this unit

The definite integral of a function gives us the area under the curve of that function. Another common interpretation is that the integral of a rate function describes the accumulation of the quantity whose rate is given. We can approximate integrals using Riemann sums, and we define definite integrals using limits of Riemann sums. The fundamental theorem of calculus ties integrals and derivatives together and can be used to evaluate various definite integrals.