Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on the following integration concepts:

• Definite integration using the Fundamental Theorem of Calculus

• Integration of polynomial and power functions

• Integration of trigonometric functions ($\sin x$, $\cos x$, ${\sec }^{2}x$)

• Integration by substitution ($u$-substitution) for composite functions

• Integration by parts for logarithmic and inverse trigonometric functions

• Partial fraction decomposition for rational functions

• Integration of exponential functions

• Algebraic simplification of integrands before integration

• Volume of Revolution (Disk Method)

• Trapezium Rule for numerical estimation of definite integrals

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Important Formulas

Fundamental Theorem of Calculus

${\int }_a^{b}f(x)dx=F(b)-F(a)\text{where }{F}^{\prime }(x)=f(x)$

Power Rule for Integration

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$

Basic Trigonometric Integrals

$\int \cos xdx=\sin x+C$

$\int {\sec }^{2}xdx=\tan x+C$

$\int \sin xdx=-\cos x+C$

Integration by Parts

$\int udv=uv-\int vdu$

Power Reduction Formula

${\cos }^{2}x=\frac{1+\cos (2x)}{2}$

Partial Fractions (distinct linear factors)

$\frac{1}{(x+a)(x+b)}=\frac{1}{b-a}\left(\frac{1}{x+a}-\frac{1}{x+b}\right)$

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Volume of Revolution — Disk Method

When a region bounded by $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$ is rotated about the $x$-axis, the volume of the resulting solid is:

$V=\pi {\int }_a^b[f(x){]}^{2}dx$

Each thin slice perpendicular to the $x$-axis is a disk of:

• Radius $=f(x)$
• Area $=\pi [f(x){]}^2$
• Thickness $=dx$

Similarly, rotating about the $y$-axis with $x=g(y)$: $V=\pi {\int }_c^d[g(y){]}^{2}dy$

Worked Example: Find the volume when $y=\sqrt{x}$ from $x=0$ to $x=4$ is rotated about the $x$-axis.

$V=\pi {\int }_0^4(\sqrt{x}{)}^{2}dx=\pi {\int }_0^{4}xdx=\pi {\left[\frac{x^2}{2}\right]}_0^4=\pi \cdot 8=8\pi \text{ cubic units}$

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Trapezium Rule

The Trapezium Rule estimates ${\int }_a^{b}f(x)dx$ by dividing $[a,b]$ into $n$ equal strips of width $h=\frac{b-a}{n}$:

${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots +2f(x_{n-1})+f(x_n)\right]$

where $x_i=a+ih$ for $i=0,1,2,\dots ,n$.

Key points:

• The first and last ordinates have coefficient 1.
• All interior ordinates have coefficient 2.
• The result is multiplied by $\frac{h}{2}$.

Worked Example: Estimate ${\int }_0^4{x}^{2}dx$ using the Trapezium Rule with $n=4$.

$h=1$; ordinates at $x=0,1,2,3,4$: $f=0,1,4,9,16$.

$\approx \frac{1}{2}[0+2(1)+2(4)+2(9)+16]=\frac{1}{2}[0+2+8+18+16]=\frac{44}{2}=22$

(Exact value: $\frac{64}{3}\approx 21.33$)

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Summary

This exercise provides comprehensive practice in evaluating definite integrals using various techniques. Key topics include:

• FTC: ${\int }_a^{b}f(x)dx=F(b)-F(a)$
• Substitution: Change limits when substituting in definite integrals
• Integration by Parts: $\int udv=uv-\int vdu$ (use LIATE rule for choosing $u$)
• Partial Fractions: Decompose rational functions before integrating
• Volume of Revolution: $V=\pi {\int }_a^b[f(x){]}^{2}dx$ (Disk Method)
• Trapezium Rule: Numerical estimation with coefficients $1,2,2,\dots ,2,1$

Key strategies: recognize when algebraic simplification is needed, apply even/odd function properties over symmetric intervals, and select the appropriate technique based on the integrand's structure.