Question

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the y-axis.
y = x^3/2,

y=8

x=0

Answer 1

Solution :

$y=x^{3/2}, \ \ y =8, \ \ x=0$

$v=2 \pi \int_0^4 x (8-x^{3/2}) \ dx$

 $= 2\pi \left(\frac{8x^2}{2}-\frac{x^{7/2}}{7/2}\ \left|_0^4$

 $=2\pi \left( 4x^2-\frac{2}{7}x^{7/2}\ |_0^4$

 $=2\pi \left( 64-\frac{2}{7} \times 32 \times 4 \right)$

  $=2\pi \left(1-\frac{4}{7} \right) 64$

  $=128 \pi \times \frac{3}{7}$

 $=\frac{384 \pi}{7}$