Question

(1 point) Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y=x^3,y=4x

Answer 1

(1664π)/7

Explanation:

We first need to know the points of intersection of the two functions, y = x³ and y = 4x.

To do this, we equate both functions

x³ = 4x

x³– 4x = 0

x(x² – 4) = 0

x(x – 2)(x + 2) = 0

x = 0, x = 2 and x = –2

Since we are looking for the volume formed by rotating the region inside the FIRST QUADRANT, then the needed points of x are 0 and 2. These are our limits of integration.

The formula for the area of the solid is

`\int\limits^2_0 {\pi (4x)^3} \, dx -\int\limits^2_0 {\pi (x^3)^3} \, dx`

This gives:

`\int\limits^2_0 {64\pi x^3} \, dx -\int\limits^2_0 {\pi x^6} \, dx`

On integration, we have

16π(2⁴) – π(2^7)/7

=

256π – 128π/7

= (1792π – 128π)/7

= (1664π)/7