How do you do this question?

9π

Step-by-step explanation:

Start by graphing the region.  The region bound by y = x, x = 3, and y = 0 is a triangle.  When rotated about the x-axis, the resulting shape is a cone.

First, let's find the volume using integration.  To do that, slice the cone into circular disks.  Each disk has a thickness of dx, and a radius of y.  So the volume of each disk is:

dV = π r² h

dV = π y² dx

y = x, so:

dV = π x² dx

The total volume is the sum of all the disks from x=0 to x=3:

V = ∫ dV

V = ∫₀³ π x² dx

Integrating:

V = π/3 x³ |₀³

V = π/3 (3)³ − π/3 (0)³

V = 9π

We can check our answer using geometry.  Volume of a cone is:

V = ⅓ π r² h

V = ⅓ π (3)² (3)

V = 9π

70.30

step-by-step explanation: