find the volume: y = 7x^2, y = 7x, x ≥ 0; about the x-axis

Answer 1

Limits of integration: 7x² = 7x; x = 0, x = 1.

By Shoulder Method V = π∫₀¹((7x)² - (7x²)²)dx = 49π(x³/3 - x⁵/5)₀¹ = 49π·2/15 = 98π/15

Answer 2

Well here is the main part of it.....

First you have to find where they meet

7x^2 - 7x = 0

7x(x-1) = 0

7x=0 or x-1=0

x=0 or x=1

So they both start at (0,0) and meet at (1,7)

The line is above the parabola on (0,1) and then the parabola is above the line forevermore.

So then:

integral of

7x-x^2 on [0,1]

integrates to:

(7/2)x^2 - (1/3)x^3

which has limit:

[(7/2)-(1/3)] - 0 =

(21 - 2)/6 =

19/6

You can now plug the result of the integral into the

appropriate volume formula