Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

Y=1/36x^2 , x = 4, y=0 about the y-axis

Answer 1

Draw a diagram.

The graph of y = (1/36)x² intersects the graph of x = 4 at the point (4, 4/9).

Take a thin horizontal slice of the region of height Δy. The slice intersects the graph of y = (1/36)x² at the point (x, y) = (x, (1/36)x²).

Rotating the slice about the y-axis yields a washer with height Δy, outer radius 4 and inner radius x.

Volume of washer = π(4)²Δy - πx²Δy. Since y = (1/36)x², x² = 36y.

So, volume of washer = π(16 - 36y)Δy.

Volume of solid = π∫_{(0 to 4/9)} [16 - 36y)dy.