What's the volume of the resulting solid?

Question

The region in the first quadrant bounded by the line 3x+2y=7 and the coordinate axes is rotated about the x-axis. What's the volume of the resulting solid?

Answer 1

The solid formed is a cone with base radius of 7/2 and a height of 7/3

V = (1/3)(7/2)²π(7/3)

V = (7/9)(49/4)π

V = (343/36)π

V ≈ 29.93239667 units 

Answer 2

Approximately picture it first as a right triangle in the first quadrant with two sides being on the coordinate axes, and the slope where the third side lies being -3/2, then imagine rotating it.

 

With a horizontal axis of revolution, using the disk method:

 

The height of the triangle at position x is r(x) = (7-3x)/2

 

Volume = V = π∫(1/4)(7-3x)² dx    where the integral is taken from x = 0 to x = 7/3