Question

write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis.y = 81 − x2, −8 ≤ x ≤ 8

Answer 1

To find the area of the surface generated by revolving the curve y = 81 - x^2 about the x-axis, use the formula for the surface area of revolution and evaluate the definite integral.

Step-by-step explanation:

To find the area of the surface generated by revolving the curve y = 81 - x^2 about the x-axis, we can use the formula for the surface area of revolution:

S = 2π ∫[a,b] f(x) √(1 + [f'(x)]^2) dx

In this case, a = -8 and b = 8. The derivative of f(x) = 81 - x^2 is f'(x) = -2x. Plugging these values into the formula, we get:

S = 2π ∫[-8,8] (81 - x^2) √(1 + (-2x)^2) dx

Now, we can evaluate this definite integral to find the area of the surface.