Let be a real number

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

a>0

be a real number. The region in the first quadrant bounded above by the line

y=ax

, below by the parabola

y=x^{2}

, and on the left by the

y

-axis is revolved about the

x

-axis, producing a solid with a hollow core. Suppose the volume of this solid is required to be exactly

10\pi

cubic units. (a) Using vertical cross-sections, set up a definite integral that represents the volume of the solid in terms of

a

. (b) Evaluate the integral obtained in part (a) to express the volume explicitly as a function of

a

. (c) Solve for the positive value of

a

that produces the specified volume. Show every algebraic step, including any factorization or substitution you employ. (d) Confirm that the value of

a

you found in part (c) is physically reasonable by analyzing the intersection point of the two bounding curves.