Properties of Definite Integrals

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Wize University Calculus 2 Textbook > Integration

Definite Integrals

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Given that both f(x)f\left(x\right)f(x) andf′(x)f'\left(x\right)f′(x) are continuous everywhere, iff(1)=10f\left(1\right)=10f(1)=10 and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12∫−31​f′(x)dx=12, find f(−3)f\left(-3\right)f(−3).

More Definite Integrals Questions:

Definite Integral

Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right)n→∞lim​i=1∑n​2i1​sin(n2i​) as a definite integral

Definite Integral

Rewrite lim⁡n→∞∑i=1n12isin⁡(2in)\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2i}\sin\left(\frac{2i}{n}\right)n→∞lim​i=1∑n​2i1​sin(n2i​) as a definite integral

Properties of Definite Integrals

Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right)f(−x)=f(x) and g(−x)=−g(x)g\left(-x\right)=-g\left(x\right)g(−x)=−g(x). If ∫−102f(x)=5\int_{-1}^02f\left(x\right)=5∫−10​2f(x)=5, the value of ∫−11[3f(x)+x⋅sin⁡x⋅g(x)]dx=\int_{-1}^1\left[3f\left(x\right)+x\cdot\sin x\cdot g\left(x\right)\right]dx=∫−11​[3f(x)+x⋅sinx⋅g(x)]dx=

Practice: Definite Integrals and Area (~F2018 Final Q31)

Practice: Definite Integrals and Area
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\displaystyle\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]n→∞lim​Σi=1n​(n1​)[3(1+ni​)3+3]?