Question

Consider the following curve y = x^3, 0 ≤ x ≤ 5. Set up an integral in terms of x that can be used to find the area of the surface S obtained by rotating the curve about the x-axis.

Options:
a. ∫₀⁵ 2πx^3 √(1 + (3x²)²) dx
b. ∫₀⁵ 2πx √(1 + (x^3)²) dx
c. ∫₀⁵ 2πx^3 √(1 + (x²)²) dx
d. ∫₀⁵ 2πx √(1 + (3x²)²) dx

Answer 1

The integral that can be used to find the area of the surface S obtained by rotating the curve y = x^3 about the x-axis is option a. ∫₀⁵ 2πx^3 √(1 + (3x²)²) dx.

Step-by-step explanation:

The correct integral to find the area of the surface S obtained by rotating the curve y = x^3 about the x-axis is option a. ∫₀⁵ 2πx^3 √(1 + (3x²)²) dx.

To set up this integral, we need to recall the formula for finding the surface area obtained by rotating a curve about the x-axis. The formula is given by ∫₀⁵ 2πy √(1 + (dy/dx)²) dx, where y is the equation of the curve and dy/dx is its derivative. Plugging in y = x^3, we get option a as the correct integral.

Therefore, the correct answer is a. ∫₀⁵ 2πx^3 √(1 + (3x²)²) dx.