Question

The base of a solid in the xy-plane is the first-quadrant region bounded by y = x and y = x^2. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Options:
a) 1/6 cubic units
b) 1/3 cubic units
c) 1/2 cubic units
d) 1 cubic unit

Answer 1

The volume of the solid formed by semicircular cross-sections in the first quadrant and bounded by the curves y = x and y = x^2 is 1/3 cubic units.

Step-by-step explanation:

To find the volume of the solid formed by semicircular cross-sections in the first quadrant and bounded by the curves y = x and y = x2, we can apply the disk method for volume calculation. The cross-sectional area at a distance x from the y-axis will be a semicircle. Knowing the diameter of the semicircle at a specific x is the y-coordinate difference between the bounding curves, we use the formula for the area of a semicircle: A = (1/2) π (radius)2. Hence, A = (1/2) π ( (x - x2)/2 )2. The volume element dV is then A dx. Integrating this expression from x = 0 to x = 1 (the intersection points of the curves), we can find the total volume, which after calculation gives V = 1/3 cubic units, making the correct answer (b).