Chapter 6

Tilted plate Calculate the fluid force on one side of a right-triangular plate with edges 3 \(\mathrm{ft}\) , 4 \(\mathrm{ft}\) , and 5 \(\mathrm{ft}\) if the plate sits at the bottom of a pool filled with water to a depth of 6 \(\mathrm{ft}\) on its 3 -ft edge and tilted at \(60^{\circ}\) to the bottom of the pool.

Use Pappus's Theorem for surface area and the fact that the surface area of a sphere of radius \(a\) is 4\(\pi a^{2}\) to find the centroid of the semicircle \(y=\sqrt{a^{2}-x^{2}}\)

Find the volume of the solid generated by revolving each region about the \(y\)-axis. The region in the first quadrant bounded above by the parabola \(y=x^{2},\) below by the \(x\)-axis, and on the right by the line \(x=2\)

As found in Exercise \(45,\) the centroid of the semicircle \(y=\sqrt{a^{2}-x^{2}}\) lies at the point \((0,2 a / \pi) .\) Find the area of the surface swept out by revolving the semicircle about the line \(y=a\)

Find the volume of the solid generated by revolving each region about the \(y\)-axis. The region in the first quadrant bounded on the left by the circle \(x^{2}+y^{2}=3,\) on the right by the line \(x=\sqrt{3},\) and above by the line \(y=\sqrt{3}\)

Find the volume of the solid generated by revolving each region about the given axis. The region in the first quadrant bounded above by the curve \(y=x^{2},\) below by the \(x\)-axis, and on the right by the line \(x=1\) about the line \(x=-1\)

The area of the region \(R\) enclosed by the semiellipse \(y=(b / a) \sqrt{a^{2}-x^{2}}\) and the \(x\) -axis is \((1 / 2) \pi a b,\) and the volume of the ellipsoid generated by revolving \(R\) about the \(x\) -axis is 4\(/ 3 ) \pi a b^{2} .\) Find the centroid of \(R .\) Notice that the location is independent of \(a .\)

A vertical rectangular plate \(a\) units long by \(b\) units wide is submerged in a fluid of weight-density \(w\) with its long edges parallel to the fluid's surface. Find the average value of the pressure along the vertical dimension of the plate. Explain your answer.

Find the volume of the solid generated by revolving each region about the given axis. The region in the second quadrant bounded above by the curve \(y=-x^{3},\) below by the \(x\)-axis, and on the left by the line \(x=-1\) about the line \(x=-2\)

As found in Example \(8,\) the centroid of the region enclosed by the \(x\) -axis and the semicircle \(y=\sqrt{a^{2}-x^{2}}\) lies at the point \((0,4 a / 3 \pi) .\) Find the volume of the solid generated by revolving this region about the line \(y=-a\) .

Find the volume of the solid generated by revolving the region bounded by \(y=\sqrt{x}\) and the lines \(y=2\) and \(x=0\) about a. the \(x\)-axis. \(\quad\) b. the \(y\)-axis. c. the line \(y=2 . \quad\) d. the line \(x=4\)

Find the volume of the solid generated by revolving the triangular region bounded by the lines \(y=2 x, y=0,\) and \(x=1\) about a. the line \(x=1 . \quad\) b. the line \(x=2\)

Find the volume of the solid generated by revolving the triangular region bounded by the lines \(y=2 x, y=0,\) and \(x=1\) about a. the line \(y=1 . \quad\) b. the line \(y=2\) c. the line \(y=-1\)

By integration, find the volume of the solid generated by revolving the triangular region with vertices \((0,0),(b, 0),(0, h)\) about a. the \(x\)-axis. \(\quad\) b. the \(y\)-axis.

The volume of a torus The disk \(x^{2}+y^{2} \leq a^{2}\) is revolved about the line \(x=b(b>a)\) to generate a solid shaped like a doughnut and called a torus. Find its volume. (Hint: \(\int_{-a}^{a} \sqrt{a^{2}-y^{2}} d y=\) \(\pi a^{2} / 2,\) since it is the area of a semicircle of radius a.)

Volume of a bowl A bowl has a shape that can be generated by revolving the graph of \(y=x^{2} / 2\) between \(y=0\) and \(y=5\) about the \(y\)-axis. a. Find the volume of the bowl. b. Related rates If we fill the bowl with water at a constant rate of 3 cubic units per second, how fast will the water level in the bowl be rising when the water is 4 units deep?

Volume of a bowl a. A hemispherical bowl of radius \(a\) contains water to a depth \(h .\) Find the volume of water in the bowl. b. Related rates Water runs into a sunken concrete hemispherical bowl of radius 5 \(\mathrm{m}\) at the rate of 0.2 \(\mathrm{m}^{3} / \mathrm{sec} .\) How fast is the water level in the bowl rising when the water is 4 \(\mathrm{m}\) deep?

Volume of a hemisphere Derive the formula \(V=(2 / 3) \pi R^{3}\) for the volume of a hemisphere of radius \(R\) by comparing its cross-sections with the cross- sections of a solid right circular cylinder of radius \(R\) and height \(R\) from which a solid right circular cone of base radius \(R\) and height \(R\) has been removed, as suggested by the accompanying figure.

Designing a wok You are designing a wok frying pan that will be shaped like a spherical bowl with handles. A bit of experimentation at home persuades you that you can get one that holds about 3L if you make it 9 \(\mathrm{cm}\) deep and give the sphere a radius of 16 \(\mathrm{cm}\) . To be sure, you picture the wok as a solid of revolution, as shown here, and calculate its volume with an integral. To the nearest cubic centimeter, what volume do you really get? (1 \(=1000 \mathrm{cm}^{3}\) )