W06 Regular

Due date: Sunday 2/22, 11:59pm

Moments and CoM

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FlatCoMMan

Find the center of mass of FlatCoMMan. Assume a constant mass density $\rho$. Use additivity of moments.

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Solution

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CoM from Simpson’s

Use Simpson’s rule (with 6 subintervals) to estimate the CoM of this region:

You will need to estimate $M_x$ and $M_y$ and $m$ with three separate integrals. You can use a calculator for your arithmetic.

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Solution

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Improper integrals

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Proper vs. improper

For each integral below, determine whether it is proper or improper, and if improper, explain why.

(a) ${\displaystyle {\int }_0^2\frac{dx}{x^{1/3}}}$ $$ (b) ${\displaystyle {\int }_0^1{e}^{-x}dx}$ $$ (c) ${\displaystyle {\int }_0^{\pi /2}\sec xdx}$

(d) ${\displaystyle {\int }_0^{\infty }\sin xdx}$ $$ (e) ${\displaystyle {\int }_1^{\infty }\ln xdx}$ $$ (f) ${\displaystyle {\int }_0^3\ln xdx}$

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Solution

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Gabriel’s Horn - Volume and surface of revolution

The curve $y=\frac{1}{x}$ for $x>1$ is rotated about the $x$-axis. The resulting shape is Gabriel’s Horn.

(a) Find the volume enclosed by the horn by evaluating a convergent improper integral.

(b) Show that the surface area of the horn is infinite by applying comparison to a $p$-integral which is divergent.

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Solution

Solutions---0130-03

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Comparison test

Use the comparison test to determine whether the integral converges:

$${\int }_1^{\infty }\frac{dx}{x^4+\sqrt{x}}$$

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Solution

Solutions---0130-01

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Computing improper integrals

For each integral below, give the limit interpretation of improper integral and then compute the limit. Based on that result, state whether the integral converges. If it converges, what is its value?

(a) ${\displaystyle {\int }_0^1\ln xdx}$ $$ (b) ${\displaystyle {\int }_{-\infty }^{+\infty }x{e}^{-x^2}dx}$ $$ (c) ${\displaystyle {\int }_1^2\frac{1}{{(x-1)}^2}dx}$

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Solution

Solutions---0130-05