Rotating a curve to find the surface area

Question

Find the area of the surface obtained my rotating the curve x=1+4y^2 about the x-axis from y = 1 to y = 2

Answer 1

For rotation about the x - axis, the surface area formula : .

For rotation about the y - axis, the surface area formula : .

Here is the answer for the curve rotating about the y - axis.

The rotating curve x = 1 + 4y² about the y - axis from y = 1 to y = 2.

Differentiate the curve with respect to y.

dx/dy = 8y

⇒  dx/dy = 8y, a = 1, and b = 2.

.

First find : .

Apply formula : .

.

.

Answer 2

Contd...............

Then find  :  .

Let,  .

If, y = 1, then .

If, y  = 2, then

du = 128 dy

.

.

Substitute those values in equation (1).

.

Therefore, the area of the surface obtained by rotating the curve x = 1 + 4y² about the y - axis from y = 1 to y = 2 is 418.07π square units.