Suppose that R is the finite region bounded by y=x, y=x+2, x=0 and x=3. Find the exact value of the volume of the object we obtain when rotating R about the x-axis.

Answer 1

Can you see that the shapes in question are represented as follows:

Where a random cross section of the solid of revolution is shown as the red "washer".

You just need to write an equation for the volume of that red washer: Volume = (area = π•r_outside² - π•r_inside²) multiplied by the thickness "dx") and then integrate it between x = 0 and x = 3.

The outside radius (r_outside) is the distance from the x-axis to the top red line which is the y-value of y = x + 2 or, in terms of "x" it is "x + 2"

The inside radius (r_inside) is the distance from the x-axis to the bottom red line which is the y-value of y = x or, in terms of "x" it is "x"

So the integral becomes:

₀∫³ (π•(x + 2)² - π•(x)²) dx

π₀∫³ ((x + 2)² - (x)²) dx

π₀∫³ ((x² + 4x + 4) - (x²)) dx

π₀∫³ (4x + 4) dx

Let me know if you need help to find the antiderivative or to evaluate the integral.