Question

Find the volumes of the solids whose bases are bounded by the circle x^2 + y^2 = 9, with the indicated cross sections taken perpendicular to the x-axis.
a) Squares

Answer 1

x ^2 + y ^2 = 9   =>   y = y(x) = ± √(9 - x ^2)

Each cross section would be a square with side length equal to the vertical distance between the upper and lower semicircles defined by y(x), which is

√(9 - x ^2) - (- √(9 - x ^2)) = 2 √(9 - x ^2)

The area of each square section is the square of this length,

(2 √(9 - x ^2)) = 4 (9 - x ^2) = 36 - 4x ^2

We get the whole solid for -3 ≤ x ≤ 3, so integrating gives a volume of

$\displaystyle\int_{-3}^3(36-4x^2)\,\mathrm dx=\boxed{144}$