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Wize University Calculus 2 Textbook > Integration

Definite Integrals

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Definite Integrals

∫abf(x)dx\displaystyle\int_a^bf\left(x\right)dx is the definite integral of the function ff on the interval [a, b]\left[a,\ b\right]
*The definite integral of a function is a number


What does it mean?
The area bounded by the curve y=f(x)y=f(x)and the xx-axis on the interval [a, b]\left[a,\ b\right] is given by
A=lim⁡n→∞∑i=1nf(xi∗)×Δx=∫abf(x) dxA=\displaystyle\lim_{n\rightarrow\infty}\sum_{i=1}^nf\left(x_i^{\ast}\right)\times\Delta x=\int_a^bf(x)\ dx




Example
Rewrite lim⁡n→∞ ∑i=1n [2n(−5+2in)6]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\ \left[\frac{2}{n}\left(-5+\frac{2i}{n}\right)^6\right] as a definite integral
  • Δx=2n →b−a=2\Delta x=\frac{2}{n}\ \to b-a=2
  • f(xi ∗)=(−5+2in)6 → a=−5, f(x)=x6f\left(x_i^{\ \ast}\right)=\left(-5+\frac{2i}{n}\right)^6\ \to\ a=-5,\ f\left(x\right)=x^6
Therefore, this can be rewritten as the definite integral ∫−5−3x6 dx\displaystyle \int_{-5}^{-3}x^6\ dx

Practice: Definite Integral and Riemann Sum

Rewrite lim⁡n→∞∑i=1n(3n)(11+3in)\displaystyle \lim_{n\rightarrow\infty}\sum_{i=1}^n\left(\frac{3}{n}\right)\left(\frac{1}{\sqrt{1+\frac{3i}{n}}}\right) as a 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) as a definite integral
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Properties of Definite Integrals

*It has all the properties of an indefinite integral
  • ∫aaf(x)dx=0\displaystyle\int_a^af\left(x\right)dx=0
  • ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\displaystyle\int_a^bf\left(x\right)dx=\int_a^cf\left(x\right)dx+\int_c^bf\left(x\right)dx
  • ∫abf(x)dx=−∫baf(x)dx\displaystyle\int_a^bf\left(x\right)dx=-\int_b^af\left(x\right)dx
  • ∫−aaf(x)dx={0if f(x) is odd2∫0af(x)dxif f(x) is even\displaystyle\int_{-a}^af\left(x\right)dx= \begin{cases} 0&\text{if }f(x)\text{ is odd}\\ \displaystyle 2\int_0^a{f(x)dx}&\text{if }f(x)\text{ is even} \end{cases}
  • If f≥gf\ge g on the interval [𝑎, 𝑏], then ∫abf(x)dx≥∫abg(x)dx\displaystyle\int_a^bf\left(x\right)dx\ge\int_a^bg\left(x\right)dx
  • ∣∫abf(x)dx∣≤∫ab∣f(x)∣dx\displaystyle\left|\int_a^bf\left(x\right)dx\right|\le\int_a^b\left|f\left(x\right)\right|dx
Example
If ∫09f(x) dx=5\displaystyle \int_0^9 f(x)\ dx=5 and ∫952f(x) dx=6\displaystyle \int_9^5 2f(x)\ dx=6, what is ∫05f(x) dx\displaystyle \int_0^5 f(x)\ dx?
Using the properties of definite integrals, we get ∫592f(x) dx=−6 → ∫59f(x) dx=−3\displaystyle \int_5^9 2f(x)\ dx=-6\ \to \ \int_5^9 f(x)\ dx=-3
We also have that ∫09f(x) dx=∫05f(x) dx+∫59f(x) dx\displaystyle \int_0^9f\left(x\right)\ dx=\int_0^5f\left(x\right)\ dx+\int_5^9f\left(x\right)\ dx
So, 5=∫05f(x) dx+(−3)\displaystyle5=\int_0^5f\left(x\right)\ dx+\left(-3\right)
Therefore, ∫05f(x) dx=8\displaystyle\int_0^5f(x)\ dx=8

Practice: Properties of Definite Integral

Evaluate ∫−π2π2xcos⁡2x sin⁡(ex2) dx\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}x\cos^2x\ \sin \left(e^{x^2}\right)\ dx.
Extra Practice
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) 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) as a definite integral
Properties of Definite Integrals
Suppose that f(−x)=f(x)f\left(-x\right)=f\left(x\right) and g(−x)=−g(x)g\left(-x\right)=-g\left(x\right). If ∫−102f(x)=5\int_{-1}^02f\left(x\right)=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=
Properties of Definite Integrals
Given that both f(x)f\left(x\right) andf′(x)f'\left(x\right) are continuous everywhere, iff(1)=10f\left(1\right)=10 and ∫−31f′(x)dx=12\displaystyle \int_{-3}^1f'\left(x\right)dx=12, find f(−3)f\left(-3\right).
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]?