1. The solid lies between planes perpendicular to the $$x$$-axis at $$x = 0$$ and $$x = 4$$. The cross-sections perpendicular to the axis on the interval $$0 \le x \le 4$$ are squares whose diagonals run from the parabola $$y = -\sqrt{x}$$ to the parabola $$y = \sqrt{x}$$.

5. The base of a solid is the region between the curve $$y = 2\sqrt{\sin(x)}$$ and the interval $$[0, \pi]$$ on the $$x$$-axis. The cross-sections perpendicular to the $$x$$-axis are

a. equilateral triangles with bases running from the $$x$$-axis to the curve as shown in the accompanying figure.

b. squares with bases running from the $$x$$-axis to the curve.

9. The solid lies between planes perpendicular to the $$y$$-axis at $$y = 0$$ and $$y = 2$$. The cross-sections perpendicular to the $$y$$-axis are circular disks with diameters running from the $$y$$-axis to the parabola $$x = 25y^2$$.

15. Find the volume of the solid generated by revolving the shaded region about the $$x$$-axis.

17. Find the volume of the solid generated by revolving the shaded region about the $$y$$-axis.

19. Find the volume of the solid generated by revolving the region bounded by $$y=x^2$$, $$y=0$$, and $$x=2$$ about the $$x$$-axis.

Revolving around the $$x$$-axis with the disk method gives us rectangles with a width of $$dx$$. The radius of the disks is $$y=x^2$$.

$$! V=\int_{x=0}^2 \pi \left(x^2\right)^2\,dx$$

$$! V=\pi\bigg[\frac{x^5}{5}\bigg]_0^2=\frac{\pi\cdot 32}{5}\,\textrm{units}^3$$

21. Find the volume of the solid generated by revolving the region bounded by $$y=\sqrt{9-x^2}$$ and $$y=0$$ about the $$x$$-axis.

23. Find the volume of the solid generated by revolving the region bounded by $$y=\sqrt{\cos(x)}$$, $$0\le x \le \pi /2$$ about the $$x$$-axis.

25. Find the volume of the solid generated by revolving the region bounded by $$y=e^{-x}$$, $$y=0$$, and $$x=1$$ about the $$x$$-axis.

31. Find the volume of the solid generated by revolving the region bounded by the region enclosed by $$x =\sqrt{5}y^2$$, $$x=0$$, $$y=-1$$, and $$y=1$$ about the $$y$$-axis.

32. Find the volume of the solid generated by revolving the region bounded by the region enclosed by $$x=y^{3/2}$$, $$x=0$$, and $$y=2$$ about the $$y$$-axis.

35. Find the volume of the solid generated by revolving the region bounded by the region enclosed by $$x=\frac{2}{\sqrt{y+1}}$$, $$x=0$$, $$y=0$$, and $$y=3$$ about the $$y$$-axis.

37. Find the volume of the solid generated by revolving the shaded region about the $$x$$-axis.

45. Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices $$(1,0)$$, $$(2,1)$$, $$(1,1)$$ about the $$y$$-axis

53. Find the volume of the solid of revolution by revolving the region bounded by the parabola $$y=x^2$$ and the line $$y=1$$ about

a. the line $$y=1$$

b. the line $$y=-1$$

c. the line $$y=2$$

54. 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

b. the $$y$$-axis

55. **The volume of a torus**: The disk $$x^2+y^2\le a^2$$ is revolved about the line $$x=b$$ ($$b\ge a$$) to generate a solid shape like a doughnut and called a torus. Find its volume. (HINT: $$\int_{-a}^{a}\sqrt{a^2-y^2}\,dy=\frac{\pi a^2}{2}$$, since it is the area of a semicircle of radius $$a$$.)

57. **Volume of a Bowl**

a. A hemispherical bowl of radius $$a$$ contains water to a depth $$h$$. Find the volume of the water in the bowl.

b. **Related Rates:** Water runs into a sunken concrete hemispherical bowl of radius 5 m at a rate of $$0.2\,\textrm{m}^3$$/sec. How fast is the water level in the bowl rising when the water in 4 m deep?

61. **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 3 L if you make it 9 cm deep and give the sphere a radius of 16 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?

($$1\,\textrm{L}=1000\,\textrm{cm}^3$$.)

From the text:

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