W01 Regular

Due date: Sunday 1/18, 11:59pm

Shells

01 $★$

02

Shells volume - set up integrals, both axes

Consider the region in the first quadrant bounded by the lines $x=0$ and $y=2$, and the curve $y=4-x^2$.

Set up integrals to find the volumes of the solids obtained by revolving this region about (i) the $x$-axis, and (ii) the $y$-axis.

(No need to evaluate the integrals in this problem.)

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Solution

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02 $★$

03

Shells volume - shells v. washers

Consider the region in the $xy$-plane, in the first quadrant, bounded by the $y$-axis on the left, by $y=8-6x^2$ on the top, and $y=2x^2$ on the bottom.

A 3D solid is given by revolving this region around the $y$-axis.

(a) Find the volume of the solid using the method of shells.

(b) Attempt to find the volume of the solid using the method of washers/disks. Why is this harder? (TWO reasons!)

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Solution

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IBP

03 $★$

03

Integration by parts - A and L

Compute the integral:

$$\int x^3\ln xdx$$

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Solution

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04

04

Integration by parts - A and E

Compute the integral:

$${\int }_0^{3}x{e}^{4x}dx$$

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Solution

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05 $★$

05

Integration by parts - A and I

Compute the integral:

$$\int {\tan }^{-1}(x)dx$$

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Solution

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06

06

Integration by parts - E and T, “breaking the circle”

Compute the integral:

$$\int e^x\sin (x)dx$$

You should perform IBP twice, find an equation, and use algebra to solve it (“breaking the circle”) for the desired integral.

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Solution

Solutions---0010-06