A region is bounded by x=y^2 and x=4 and y=0 and revolved about the line x=5. Find the volume using shell method.

If you draw the bounded region in the x,y-plane, you'll find it to be somewhat ambiguous, but since y = 0 cuts the area between the parabola x = y ² and x = 4 perfectly in half, you can use either the top or bottom half. I'll use the top one, i.e. assume y ≥ 0.

For every x taken from the interval [0, 4], we can get a shell with height √x. The distance from x to the axis of revolution, x = 5, is 5 - x, which corresponds to the radius of the shell. The area of this shell is

2π (radius) (height) = 2π (5 - x) √x

Then the volume of the solid is the sum of infinitely many such shells made at every 0 ≤ x ≤ 4, given by the integral

add an image

step-by-step explanation:

shift 4 units up, and 1 unit left.

it will translate f(x) = 3^x to g(x) = 3^(x+1) + 4