数学代写|Math 152 Lab 3

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 $y=3$

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).

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.

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.