W01 Stepwise

Due date: Thursday 1/15, 11:59pm

Shells

01

01

Shells volume - offset graph, $y$-axis

Consider the region in the first quadrant bounded by the lines $x=0$, $x=2$, $y=0$, and the curve $y=\frac{1}{\sqrt{x^2+1}}$. Revolve this about the $y$-axis.

Find the volume of the resulting solid.

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Solution

Solutions - 0030-01

(1) Set up integral.

$$\begin{array}{cc}V& ={\int }_a^{b}2\pi Rhdr\\ & \\ & \gg \gg {\int }_0^{2}2\pi x\left(\frac{1}{\sqrt{x^2+1}}\right)dx\end{array}$$

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(2) Perform $u$-substitution with $u=x^2+1$ and $du=2x$:

$$\begin{array}{c}\gg \gg V={\int }_0^5\frac{\pi }{\sqrt{u}}du\end{array}$$

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(3) Integrate with power rule:

$$\begin{array}{c}\gg \gg \pi {\frac{u^{1/2}}{1/2}|}_0^5\\ \\ \gg \gg 2\pi (\sqrt{5}-1)\end{array}$$Link to original

IBP

02

01

Integration by parts - A and E

Compute the integral:

$$\int (2x+9)e^{x}dx$$

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Solution

Solutions - 0010-01

(1) Select $u$ and $v^{\prime }$ considering LIATE:

$$\begin{array}{cc}u=2x+9& v^{\prime }=e^x\\ u^{\prime }=2& v=e^x\end{array}\begin{array}{c}⟶\int u\cdot v^{\prime }\text{original}\\ ⟶\int u^{\prime }\cdot v\text{final}\end{array}$$

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(2) Apply IBP formula $uv-\int u^{\prime }\cdot v$:

$$\begin{array}{cccc}& uv-\int u^{\prime }\cdot v& \gg \gg (2x+9)e^x-\int 2e^{x}dx& \\ & & & \\ & & \gg \gg (2x+9)e^x-(2e^x+C)& \\ & & & \\ & & \gg \gg (2x+7)e^x+C& \text{(Note A)}\end{array}$$

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Note A: We can change notation $-C$ to $+C$ because the value of $C$ is arbitrary.

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03

02

Integration by parts - A and T

Compute the integral:

$$\int x^2\sin xdx$$

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Solution

Solutions - 0010-02

(1) Select $u$ and $v^{\prime }$ considering LIATE:

$$\begin{array}{cc}u=x^2& v^{\prime }=\sin x\\ u^{\prime }=2x& v=-\cos x\end{array}\begin{array}{c}⟶\int u\cdot v^{\prime }\text{original}\\ ⟶\int u^{\prime }\cdot v\text{final}\end{array}$$

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(2) Apply IBP formula $uv-\int u^{\prime }\cdot v$:

$$\begin{array}{cccc}& uv-\int u^{\prime }\cdot v& \gg \gg x^2(-\cos x)-\int 2x(-\cos x)dx& \\ & & & \\ & & \gg \gg -x^2\cos x+2\int x\cos xdx& \text{(Exp. A)}\end{array}$$

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(3) Select another $u=x$ and $v^{\prime }=\cos x$ and do IBP again:

$$\begin{array}{cc}\int x\cos x& \gg \gg x\sin x-\int \sin xdx\\ & \\ & \gg \gg x\sin x+\cos x+C\end{array}$$

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(4) Put all together in (A):

$$\begin{array}{ccc}& \gg \gg -x^2\cos x+2(x\sin x+\cos x+C)& \\ & & \\ & \gg \gg -x^2\cos x+2x\sin x+2\cos x+C& \text{(Note B)}\end{array}$$

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Note B: We can change notation $2C$ to $C$ because the value of $C$ is arbitrary.

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