Question

The region in the first quadrant bounded above by the line yequals=77​, below by the curve yequals=startroot 7 x endroot7x​, and on the left by the ​y-axis is revolved about the line yequals=77. find the volume of the resulting solid.

Answer 1

The differential of volume can be a disk of radius (7-√(7x)) and thickness dx, so is

... dV = π·r²·dx = π(7-√(7x))²·dx

Integrated over the region 0 ≤ x ≤ 7, this becomes

`V=\displaystyle \int_(0)^(7){\pi\left(7-√(7x)\right)^(2)\,dx}\\\\=\pi\left[49x-(28x√(7x))/(3)+(7x^2)/(2)\right]\limits_(0)^(7)\\\\=49\pi\left(7-(28)/(3)+(7)/(2)\right)\\\\V=343\cdot (\pi)/(6)\approx 179.59438`