21. \( y=x^{2}, x=y^{2} ; \quad \) about \( y=1 \) 22. \( y=x^{3}, y=1, x=2 ; \quad \) about \( y=-3 \) 23. \( y=3, y=1+\sec x,-\pi / 3 \leqslant x \leqslant \pi / 3 ; \quad \) about \( y=1 \) 24. \( y=\sin x, y=\cos x, 0 \leqslant x \leqslant \pi / 4 ; \quad \) about \( y=-1 \) 25. \( y=x^{3}, y=0, x=1 ; \quad \) about \( x=2 \) 26. \( x y=1, y=0, x=1, x=2 ; \quad \) about \( x=-1 \) 27. \( x=y^{2}, x=1-y^{2} ; \quad \) about \( x=3 \) 28. \( y=x, y=0, x=2, x=4 ; \quad \) about \( x=1 \)

To find the volumes of the solids of revolution for the given problems, we will use the method of cylindrical shells or the disk/washer method, depending on the situation. Below are the solutions for each problem: ### 21. \( y=x^{2}, x=y^{2} \) about \( y=1 \) 1. **Find the intersection points**: Set \( y = x^2 \) and \( x = y^2 \). This gives \( y = (y^2)^2 = y^4 \), leading to \( y(y^3 - 1) = 0 \). Thus, \( y = 0 \) or \( y = 1 \). The points of intersection are \( (0,0) \) and \( (1,1) \). 2. **Volume using the washer method**: The outer radius is \( R = 1 - y^2 \) and the inner radius is \( r = 1 - x^2 \). The volume \( V \) is given by: \[ V = \pi \int_0^1 \left( (1 - y^2)^2 - (1 - x^2)^2 \right) dy \] Convert \( x \) in terms of \( y \) and integrate. ### 22. \( y=x^{3}, y=1, x=2 \) about \( y=-3 \) 1. **Volume using the washer method**: The outer radius is \( R = 1 - (-3) = 4 \) and the inner radius is \( r = x^3 - (-3) = x^3 + 3 \). The volume \( V \) is given by: \[ V = \pi \int_0^2 \left( 4^2 - (x^3 + 3)^2 \right) dx \] ### 23. \( y=3, y=1+\sec x, -\pi / 3 \leqslant x \leqslant \pi / 3 \) about \( y=1 \) 1. **Volume using the washer method**: The outer radius is \( R = 3 - 1 = 2 \) and the inner radius is \( r = (1 + \sec x) - 1 = \sec x \). The volume \( V \) is given by: \[ V = \pi \int_{-\pi/3}^{\pi/3} \left( 2^2 - (\sec x)^2 \right) dx \] ### 24. \( y=\sin x, y=\cos x, 0 \leqslant x \leqslant \pi / 4 \) about \( y=-1 \) 1. **Volume using the washer method**: The outer radius is \( R = \sin x + 1 \) and the inner radius is \( r = \cos x + 1 \). The volume \( V \) is given by: \[ V = \pi \int_0^{\pi/4} \left( (\sin x + 1)^2 - (\cos x + 1)^2 \right) dx \] ### 25. \( y=x^{3}, y=0, x=1 \) about \( x=2 \) 1. **Volume using the shell method**: The height is \( h = x^3 \) and the radius is \( r = 2 - x \). The volume \( V \) is given by: \[ V = 2\pi \int_0^1 (2 - x)(x^3) dx \] ### 26. \( xy=1, y=0, x=1, x=2 \) about \( x=-1 \) 1. **Volume using the shell method**: The height is \( h = \frac{1}{x} \) and the radius is \( r = x + 1 \). The volume \( V \) is given by: \[ V = 2\pi \int_1^2 (x + 1) \left( \frac{1}{x} \right) dx \] ### 27. \( x=y^{2}, x=1-y^{2} \) about \( x=3 \) 1. **Find the intersection points**: Set \( y^2 = 1 - y^2 \) leading to \( 2y^2 = 1 \) or \( y = \pm \frac{1}{\sqrt{2}} \). 2. **Volume using the shell method**: The height is