Use the general slicing method to find the volume of the following solid. The solid whose base is the triangle with vertices (0,0), (16,0), and (0,16) and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles. Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A.
B.
The Correct Answer and Explanation is:
The correct choice is A.
The integral is:
V = ∫ from 0 to 16 of (π/8)(16-x)² dx
Explanation:
- Determine the Variable of Integration:
The problem specifies that the cross-sections are “parallel to the y-axis.” This means the slices are vertical, standing up from the triangular base. The thickness of each slice is an infinitesimal change along the x-axis, represented as dx. Therefore, the integration must be performed with respect to x. This leads to selecting option A. - Find the Limits of Integration:
The base of the solid is a triangle in the xy-plane with vertices (0,0), (16,0), and (0,16). This triangular base spans the x-axis from x = 0 to x = 16. These values serve as the lower and upper limits of integration. - Define the Cross-Section’s Dimensions:
For any given x-value, the slice is a semicircle. The base of this semicircle is a vertical line segment within the triangle, extending from the x-axis (where y=0) up to the hypotenuse of the triangle. The hypotenuse connects the points (16,0) and (0,16), and its equation is y = 16 – x. The length of this base segment is s = y – 0 = 16 – x. This length s is the diameter of the semicircular cross-section. - Formulate the Area Function A(x):
The area of a semicircle is given by the formula A = (1/2)πr², where r is the radius. The radius is half the diameter, so r = s/2 = (16 – x)/2.
Substituting the radius into the area formula gives the area of a single cross-section as a function of x:
A(x) = (1/2) * π * [ (16 – x) / 2 ]²
A(x) = (1/2) * π * ( (16 – x)² / 4 )
A(x) = (π/8)(16 – x)² - Set Up the Integral:
The volume of the solid is found by integrating the cross-sectional area function A(x) over the interval defined by the limits of integration.
The resulting integral is:
Volume = ∫₀¹⁶ A(x) dx = ∫₀¹⁶ (π/8)(16 – x)² dx
