Question

Find the volume

Answer 1

The curves are y = x³ and y = 25x.

First we find out the points where the curve touch each other

x³ - 25x = 0

x(x²-25) = 0

x = 0, x² = 25 ⇒ x = ± 5.

Since the volume of the region is to calculated in fisrt quadrant then x = 0, x = 5

If x = 0 then y = 0

If x = 5 then y = 125

Therefore the curves touch each other at (0,0) and (5,125).

Graph

Area of the region bounded by the curves along x - axis is πy² ⇒ π((25x)² - (x³)²) ⇒ π(625x² - x^6)

Volume of the region bounded by the curve is .

V(x) = 14880.95π

Therefore the volume of the solid is 14880.95π.