Question

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the y-axis. (Round your final answer to three decimal places.)

y = 1 − x2/16 , 0 ≤ x ≤ 4

Answer 1

To find the area of the surface generated by revolving the curve y = 1 − x^2/16 about the y-axis, we can use the formula for finding the surface area of a curve rotated about the y-axis. We substitute the values into the formula and evaluate the integral to find the surface area.

Step-by-step explanation:

To find the area of the surface generated by revolving the curve y = 1 − x2/16 about the y-axis, we can use the formula for finding the surface area of a curve rotated about the y-axis. The formula is:

Surface Area = 2π∫(f(x) * √(1 + (f'(x))2)) dx

In this case, f(x) = 1 − x2/16 and f'(x) = -x/8. We need to evaluate the integral from x = 0 to x = 4.

Substituting the values into the formula and evaluating the integral, we get:

Surface Area = 2π∫(1 − x2/16) * √(1 + (-x/8)2) dx

= 2π∫(1 − x2/16) * √(1 + x2/64) dx

Now, we can integrate using a calculator or computer software to get the final answer.