Question

Use the region in the first quadrant bounded by √x, y=2 and the y-axis to determine the volume when the region is revolved around the line y = -2. Evaluate the integral.

A. 18.667
B. 17.97
C. 58.643
D. 150.796
E. 21.333
F. 32.436
G. 103.323
H. 27.4

Answer 1

To determine the volume when the region is revolved around the line y = -2, we can use the shell method. We need to integrate the circumference of a shell multiplied by its height.

The circumference of a shell with radius r and height h is given by 2πr, and the height of each shell is given by y + 2.

The first quadrant bounded by √x, y = 2 and the y-axis creates a solid that is symmetrical about y axis. We can integrate from y = 0 to y = 2 to obtain the volume of the solid.

The integral becomes:

V = ∫(2πy)((√y+2)^2)dy

After simplification, we get:

V = 32π/5 + 128π/3

The value of V is approximately 103.323

Therefore, the correct answer is (G) 103.323.