Chapter 28

A tent has a base that is an isosceles triangle. The mouth of the tent measures 8 feet and the length is 12 feet. The tent is constructed so that the cross sections perpendicular to the base are all equilateral triangles. Find the volume of the tent.

A Wisconsin cheese factory makes its cheese in solid cylinders of radius 2 inches. A wedge of cheese is cut from the cylinder by chopping through the diameter of the base at an angle of 45 degrees with the base. Find the volume of the wedge of cheese.

Approximate each length with error less than 0.05. The length of one arc of the cosine curve, say from \(x=\frac{-\pi}{2}\) to \(x=\frac{\pi}{2}\).

Find the volume generated when the region in the first quadrant bounded by \(y=x^{2}\) and \(y=3 x\) is rotated about (a) the \(x\) -axis, (b) the \(y\) -axis, (c) the line \(x=-1\).

Find the volume generated when the region bounded by the \(y\) -axis, \(y=x^{2}\), and \(y=4\) is rotated about the \(x\) -axis. Do this in three ways. (a) Chop the shaded region into vertical strips and rotate. (b) Chop the shaded region into horizontal strips and rotate. (c) Subtract volumes. Subtract the volume generated by rotating the region under \(y=x^{2}\) from that generated by rotating the region under \(y=4\). (The latter is just a cylinder.)

Find the volume generated by rotating the region bounded by the \(x\) -axis and \(y=\sqrt{\sin x}\) for \(0 \leq x \leq \pi\) about the line \(y=0\).

Find the work done in pushing a stroller 200 feet along a path by applying a constant force of 12 pounds in the direction of motion.

The region bounded above by \(y=\frac{1}{x}\), below by the \(x\) -axis, and laterally by \(x=\frac{1}{2}\) and \(x=5\) is rotated about the \(x\) -axis. Find the volume of the funnel generated.

A force of 5 pounds will stretch a certain spring 3 inches beyond its natural length. (a) What is the value of the spring constant? (b) How much work is done in stretching the spring from its natural length to 3 inches beyond its natural length? (c) How much work is done in stretching the spring from 3 inches beyond its natural length to 6 inches beyond its natural length? (d) Why is the answer to part (c) larger than the answer to part (b)?

Let \(A\) be the region bounded by \(y=x^{2}\) and \(y=4-x^{2} .\) Find the volume generated by rotating region \(A\) about (a) the \(y\) -axis, (b) the \(x\) -axis.

Suppose \(1.2\) foot-pounds of work are required to compress a spring 2 inches. (a) How much work is required to stretch this spring 2 inches from its equilibrium position? (b) How much work is required to stretch this spring 4 inches from equilibrium? (c) How much work is required to stretch the spring 5 inches from its equilibrium position?

A window washer weighing 160 pounds is attached to a rope hanging from the roof of the building whose windows he is washing. The rope weighs \(0.6 \mathrm{lb} / \mathrm{ft}\). Right now he is working 50 feet down from the rooftop. (a) How much work is required to bring him to the windows that are 25 feet from the rooftop? (b) How much work will it take to bring him from where he is to the roof?

Find the volume generated by revolving the region bounded by \(y=x^{2}\) and \(y=4\) about (a) the \(y\) -axis, (b) the vertical line \(x=2\), (c) the horizontal line \(y=4\), (d) the horizontal line \(y=5\).

Amelia and Beulah are city dwellers who have set up pulley systems to get their groceries delivered without walking the stairs. Amelia pulls her basket filled with 12 pounds of groceries up to her 40 -foot-high balcony. Beulah pulls her basket filled with 16 pounds of cleaning supplies up to her 30 -foot-high window. Assuming both women use ropes weighing \(0.2 \mathrm{lb} / \mathrm{ft}\), whose task requires more work? How much more work? (Assume friction is negligible.)

A parfait cup is formed by revolving the curve \(y=x^{3}, 0 \leq x \leq 2\), about the \(y\) -axis. The parfait cup is filled to the brim with hot chocolate. If you plan to drink exactly half the hot chocolate in the cup, at what height should the liquid be when you stop drinking?

A hemispherical tank with a radius of 7 feet is filled to a height of 6 feet with gasoline. How much work is required to pump all the gasoline over the top? The weight-density of gasoline is \(42 \mathrm{lb} / \mathrm{ft}^{3}\).

We model a large, 9 -inch-tall soup bowl by rotating a region \(A\) around the \(y\) -axis. \(A\) is the region bounded by the \(y\) -axis, \(y=\frac{1}{4} x^{2}\), and \(y=9\). (a) What is the capacity of the bowl? (b) If the bowl was filled to the brim when set out on the table and an hour later was filled only to a height of 4 inches, how much soup was ladled out in the hour?

Find the volume of the ellipsoid generated by revolving the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) about the \(x\) -axis.

Pick an application of integration not explicitly discussed in this text and learn about it. Then either write an exposition of the application or prepare a short lesson on it. Include examples.