Question

The area is bounded by the curve y=x^2+2 and the line y=x+8 is rotated around the x axis. Find the volume of the solid generated.

Answer 1

Described area:

Then

$V = \pi \int\limits_a^b {\left( {{f^2}(x) - {g^2}(x)} \right)dx} = \pi \int\limits_{ - 2}^3 {({{(x + 8)}^2} - {{({x^2} + 2)}^2}} )dx =$

$= \pi \int\limits_{ - 2}^3 {\left( {{x^2} + 16x + 64 - {x^4} - 4{x^2} - 4} \right)dx = } \pi \int\limits_{ - 2}^3 {\left( { - {x^4} - 3{x^2} + 16x + 60} \right)dx = }$

$= \pi \left( { - \left. {\frac{{{x^5}}}{5}} \right|_{ - 2}^3 - \left. {{x^3}} \right|_{ - 2}^3 + \left. {8{x^2}} \right|_{ - 2}^3 + 60\left. x \right|_{ - 2}^3} \right) =$

$= \pi \left( { - \frac{{243 + 32}}{5} - (27 + 8) + 8(9 - 4) + 60(3 + 2)} \right) = \pi \left( { - 55 - 35 + 40 + 300} \right) = 250\pi$