Simpson's Rules for the area of a surface around the x-axis

Question

Use Simpson's Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by your calculator. (Round your answer to six decimal places.)

y = sec x,

 

0 ≤ x ≤ π/6

 

Answer 1

Step 1:

The function is and .

Definition of surface area:

If the curve is described as ,  then the surface area of the curve obtained by rotating about the -axis is

The curve is .

Differentiate on each side.

Area of the surface obtained by rotating the curve about the -axis is

Step 2:

Simpson's Rule:

Let  be continuous on . The Midpoint Rule for approximating  is given by

,

where  and

Using Simpson Rule,

Step 3:

Area of the surface obtained by rotating the curve about the -axis is

Using calculator, the value of the integral is

Solution:

Area of the surface obtained by rotating the curve about the -axis is