Problem 1.

1. Let $R$ be the region bounded by $y=x$ and $y=x e^{1-x / 2}$.
(a) Sketch the region $R$ and the line $y=3$.
(b) Find the exact and approximate volume of the solid formed by rotating $R$ about the $y$ -axis.
(c) Find the exact and approximate volume of the solid formed by rotating $R$ about the line $y=3$

Problem 2.

A spherical tank is $3 \mathrm{~m}$ tall and therefore has a capacity of about $14000 \mathrm{~L}$ (NOTE that $3 \mathrm{~m}$ is its diameter, and one cubic meter is $1000 \mathrm{~L}$ ). You have been asked to put level lines on the tank showing the depths where the tank contains 1000,5000 , and $10000 \mathrm{~L}$ of liquid.
(a) Use calculus to determine the height of each line (to the nearest $\mathrm{cm}$ ).
(b) Plot the circular cross-section of the tank and the level lines (NOTE the lines do not have to stay within the circle).

Problem 3.

An elevator weighing 800 lbs is suspended by a 120 foot cable that weighs 10 lb per foot.
(a) Calculate the work done in moving the elevator from the ground floor to a height of 30
feet.
(b) Calculate the work done in moving the elevator from the ground floor to the top, i.e.,
the full 120 feet.

Problem 4.

4. Suppose the tank in $\# 2$ is full of water (weight density $9800 \mathrm{~N}$ ). Find the work required to pump all the water out of the tank…
(a) …if the tank is full.
(b) $-(\mathrm{d})$… if the tank is at each of the level lines calculated in #2.

Math 152的题目并不算难做，只要上课听懂，做起来准确率和效率都会时非常高的，如果实在需要帮助，也欢迎联系我们。

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