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# Unit: 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)

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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)

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)

### Learn

### 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!

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)

### Learn

### 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!

Level up on the above skills and collect up to 560 Mastery points

AP Calc:

FUN (BI)

, FUN‑5 (EU)

, FUN‑5.A (LO)

, FUN‑5.A.1 (EK)

, FUN‑5.A.2 (EK)

### Learn

### 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!

AP Calc:

FUN (BI)

, FUN‑5 (EU)

, FUN‑5.A (LO)

, FUN‑5.A.3 (EK)

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.A (LO)

, FUN‑6.A.1 (EK)

, FUN‑6.A.2 (EK)

### Learn

### 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!

Level up on the above skills and collect up to 560 Mastery points

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)

### Learn

### 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!

Level up on the above skills and collect up to 160 Mastery points

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.C (LO)

, FUN‑6.C.1 (EK)

, FUN‑6.C.2 (EK)

### Learn

### 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!

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.C (LO)

, FUN‑6.C.1 (EK)

, FUN‑6.C.2 (EK)

### Learn

### Practice

- Indefinite integrals: eˣ & 1/xGet 3 of 4 questions to level up!
- Indefinite integrals: sin & cosGet 3 of 4 questions to level up!

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.B (LO)

, FUN‑6.C (LO)

### Learn

### 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!

Level up on the above skills and collect up to 720 Mastery points

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.D (LO)

, FUN‑6.D.1 (EK)

, FUN‑6.D.2 (EK)

### Learn

### 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!

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.D (LO)

, FUN‑6.D.3 (EK)

### Learn

### Practice

- Integration using long divisionGet 3 of 4 questions to level up!
- Integration using completing the squareGet 3 of 4 questions to level up!

Level up on the above skills and collect up to 400 Mastery points

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.E (LO)

, FUN‑6.E.1 (EK)

### Learn

### Practice

- Integration by partsGet 3 of 4 questions to level up!
- Integration by parts: definite integralsGet 3 of 4 questions to level up!

AP Calc:

FUN (BI)

, FUN‑6 (EU)

, FUN‑6.F (LO)

, FUN‑6.F.1 (EK)

### Practice

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

AP Calc:

LIM (BI)

, LIM‑6 (EU)

, LIM‑6.A (LO)

, LIM‑6.A.1 (EK)

, LIM‑6.A.2 (EK)

### Learn

### Practice

- Improper integralsGet 3 of 4 questions to level up!

Level up on the above skills and collect up to 320 Mastery points

Up next for you:

Level up on all the skills in this unit and collect up to 3400 Mastery points!#### Unit test

### 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.AP® is a registered trademark of the College Board, which has not reviewed this resource.