Chapter 6 - Applications of the Integral - 6.3 Volumes of Revolution - Exercises - Page 305: 25

$32 \pi$

$\ Volume, V=\pi \int_{m}^n R^2 \ dy =(\pi) \int_{-12}^4 (\dfrac{y+12}{8})^2+\pi \int_{4}^{12} (\dfrac{12-y}{4})^2 \\=\dfrac{\pi}{64} [\int_{-12}^4 (y^2+24y+144)] \ dy +\int_{4}^{12} (144-24y+y^2) \ dy] \\=\dfrac{\pi}{64} [(\dfrac{1}{3} y^3 +12y^2 +144 y )_{-12}^4 + [\dfrac{1}{3} y^3 -12y^2 +144 y ]_{4}^{12} ]\\=\dfrac{64 \pi}{3}+\dfrac{ 32 \pi}{3} \\=32 \pi$