Volume by Disks for Rotation About the $x$-Axis
$V = \int_{a}^{b} A(x) dx = \int_{a}^{b} \pi [R(x)]^2 dx.$
Volume by Disks for Rotation About the $y$-Axis
$V = \int_{c}^{d} A(y) dy = \int_{c}^{d} \pi [R(y)]^2 dy.$
By using the above formulas, solve the following problems:
1. Calculate the volumes of the solids generated by revolving the regions bounded by the lines and the curves about the $x$-axis.
a) $y = x^2, y = 0, x = 2$
b) $y = x - x^2, y = 0$
c) $y = \sqrt{\cos x}, 0 \le x \le \pi/2, y = 0, x = 0$
2. Calculate the volumes of the solids generated by revolving the regions bounded by the lines and the curves about the $y$-axis.
a) $x = \sqrt{5y^2}, x = 0, y = -1, y = 1$
b) $x = \sqrt{2 \sin 2y}, 0 \le y \le \pi/2, x = 0$
c) $x = 2/\sqrt{y+1}, x = 0, y = 0, y = 3$
