Find the volume of the solid generated by revolving the region bounded by y = sqrt(x), the x-axis, and x = 4 around the x-axis.
Answer and explanation
Correct answer: B. 8 * pi
- Step 1: Use the disk method: V = pi * integral from a to b of [f(x)]^2 dx.
- Step 2: Here, f(x) = sqrt(x), so [f(x)]^2 = x. The bounds are from x = 0 to x = 4.
- Step 3: Evaluate: V = pi integral from 0 to 4 of x dx = pi [x^2 / 2] from 0 to 4 = pi (16 / 2) = 8 pi.
Answer
8 * pi.
Why not the others: '16 pi' fails to divide by 2 during integration; '4 pi' divides 16 by 4; '32 * pi / 3' mistakenly integrates x^2 instead of [sqrt(x)]^2.
Practise more Calculus and Mathematical Analysis.
Free questions with an explanation for every one, and full timed General Secondary Certificate (Tawjihiya Kuwait) mock tests.
More Calculus and Mathematical Analysis questions
- Evaluate the trigonometric limit: lim (x -> 0) [sin(5x) / (3x)].
- Find the derivative of the rational function f(x) = (2x - 3) / (x + 4) for x != -4.
- What is the slope of the tangent line to the curve y = x^3 - 3x^2 + 2 at the point where x = 3?
- Given the function f(x) = 2x^3 - 9x^2 + 12x - 5, find the x-coordinates of all local extrema (critical points).
- Determine the point of inflection for the cubic curve f(x) = x^3 - 6x^2 + 9x + 1.
- Evaluate the indefinite integral: integral of (6x^2 - 4x + 5) dx.