Exam 1 Prep: Area, Volume and Work

← All MATH 116 worksheets · LaTeX source

I. Area

1. Find the area of the region enclosed by \(x=y^4\), \(y=\sqrt{2-x}\) and \(y=0\).
2. Find the area of the region bounded by \(y=\tan x\) and \(y=2\sin x\) on \(-\tfrac{\pi}{3}\le x\le \tfrac{\pi}{3}\).
3. Find the number \(b\) such that the line \(y=b\) divides the region bounded by \(y=x^2\) and \(y=4\) into two regions of equal area.

II. Volume

1. The region bounded by \(y=x^2\) and \(x=y^2\) is rotated about \(y=1\). Find the volume of the resulting solid.
2. The region bounded by \(x=y^2\) and \(x=1-y^2\) is rotated about \(x=3\). Set up the integral for the volume of the resulting solid.

III. Work

1. If 6 J of work is needed to stretch a spring from 10 cm to 12 cm, and another 10 J from 12 cm to 14 cm, find the work required to stretch the spring from 9.5 cm to 13.5 cm.
2. A 10 kg bucket is hoisted at constant speed from the ground to a platform 12 m high. The rope passes over a light pulley at the platform and has linear density 0.8 kg/m. The bucket starts with 36 kg of water, which leaks at a constant rate and is completely gone by the time the bucket reaches 8 m. Compute the total work done by the lifting force.

IV. Integration

1. \[\displaystyle \int x\arcsin x\,dx\]
2. \[\displaystyle \int \frac{x^{2}}{\sqrt{x^{2}-4}}\,dx\]