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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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About this unit
The definite integral is an important tool in calculus. It calculates the area under a curve, or the accumulation of a quantity over time. Riemann sums allow us to approximate integrals, while the fundamental theorem of calculus reveals how they connect to derivatives.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!
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!
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!
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!
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!
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!
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