Final answer:
To find the exact area of the surface obtained by rotating the curve y = x^(3/2) about the x-axis, you can use the formula for the surface area of a solid of revolution. The exact area of the surface is (4π/5) × (a^5), where a is the maximum value of x.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve y = x^(3/2) about the x-axis, we can use the formula for the surface area of a solid of revolution. The formula is given as: Area = 2π ∫ y √(1 + (dy/dx)^2) dx.
In this case, we have y = x^(3/2). Differentiating with respect to x gives us dy/dx = (3/2)x^(1/2). Substituting these values into the formula and performing the integration gives us the exact area of the surface.
By evaluating the integral, we find that the exact area of the surface is (4π/5) × (a^5), where a is the maximum value of x.