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Integration and accumulation of change

AP Calc: CHA (BI), CHA‑4 (EU), FUN (BI), FUN‑5 (EU), FUN‑6 (EU), LIM (BI), LIM‑5 (EU), LIM‑6 (EU)

Exploring accumulations of change

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑4.A (LO), CHA‑4.A.1 (EK), CHA‑4.A.2 (EK), CHA‑4.A.3 (EK), CHA‑4.A.4 (EK)
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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!

Approximating areas with Riemann sums

AP Calc: LIM (BI), LIM‑5 (EU), LIM‑5.A (LO), LIM‑5.A.1 (EK), LIM‑5.A.2 (EK), LIM‑5.A.3 (EK), LIM‑5.A.4 (EK)
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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
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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!

Riemann sums, summation notation, and definite integral notation

AP Calc: LIM (BI), LIM‑5 (EU), LIM‑5.B (LO), LIM‑5.B.1 (EK), LIM‑5.B.2 (EK), LIM‑5.C (LO), LIM‑5.C.1 (EK), LIM‑5.C.2 (EK)
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Summation notation
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Summation notation
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Worked examples: 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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Definite integral as the limit of a Riemann sum
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Definite integral as the limit of a Riemann sum
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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
Summation notationGet 3 of 4 questions to level up!
Riemann sums in summation notationGet 3 of 4 questions to level up!
Definite integral as the limit of a Riemann sumGet 3 of 4 questions to level up!

The fundamental theorem of calculus and accumulation functions

AP Calc: FUN (BI), FUN‑5 (EU), FUN‑5.A (LO), FUN‑5.A.1 (EK), FUN‑5.A.2 (EK)
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The fundamental theorem of calculus and accumulation functions
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Functions defined by definite integrals (accumulation functions)
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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 involving area

AP Calc: FUN (BI), FUN‑5 (EU), FUN‑5.A (LO), FUN‑5.A.3 (EK)
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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!

Applying properties of definite integrals

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.A (LO), FUN‑6.A.1 (EK), FUN‑6.A.2 (EK)
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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
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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!

The fundamental theorem of calculus and definite integrals

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.B (LO), FUN‑6.B.1 (EK), FUN‑6.B.2 (EK), FUN‑6.B.3 (EK)
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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!

Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.C (LO), FUN‑6.C.1 (EK), FUN‑6.C.2 (EK)
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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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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!

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.C (LO), FUN‑6.C.1 (EK), FUN‑6.C.2 (EK)
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Indefinite integral of 1/x
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Indefinite integrals of sin(x), cos(x), and eˣ
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Common integrals review
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Practice
Indefinite integrals: eˣ & 1/xGet 3 of 4 questions to level up!
Indefinite integrals: sin & cosGet 3 of 4 questions to level up!

Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.B (LO), FUN‑6.C (LO)
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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!

Integrating using substitution

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.D (LO), FUN‑6.D.1 (EK), FUN‑6.D.2 (EK)
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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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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 functions using long division and completing the square

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.D (LO), FUN‑6.D.3 (EK)
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Integration using long division
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Integration using completing the square and the derivative of arctan(x)
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Practice
Integration using long divisionGet 3 of 4 questions to level up!
Integration using completing the squareGet 3 of 4 questions to level up!

Using integration by parts

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.E (LO), FUN‑6.E.1 (EK)
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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
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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

AP Calc: FUN (BI), FUN‑6 (EU), FUN‑6.F (LO), FUN‑6.F.1 (EK)
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Integration with partial fractions
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Practice
Integration with partial fractionsGet 3 of 4 questions to level up!

Evaluating improper integrals

AP Calc: LIM (BI), LIM‑6 (EU), LIM‑6.A (LO), LIM‑6.A.1 (EK), LIM‑6.A.2 (EK)
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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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Practice
Improper integralsGet 3 of 4 questions to level up!

Optional videos

Learn
Proof of fundamental theorem of calculus
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Intuition for second part of fundamental theorem of calculus
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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.