Home / AP Calculus AB 8.8 Volumes with Cross Sections: Triangles and Semicircles – MCQs

AP Calculus AB 8.8 Volumes with Cross Sections: Triangles and Semicircles - MCQs - Exam Style Questions

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Calc-Ok Question


Let \(R\) be the region in the first quadrant bounded by \(y=4\cos\!\big(\tfrac{\pi x}{4}\big)\) and \(y=(x-2)^{2}\). The region \(R\) is the base of a solid. For the solid, each cross section perpendicular to the \(x\)-axis is an isosceles right triangle with a leg in region \(R\). What is the volume of the solid?

(A) \(1.775\)
(B) \(3.549\)
(C) \(4.800\)
(D) \(5.575\)

▶️ Answer/Explanation

Intersections: \(4\cos\!\big(\tfrac{\pi x}{4}\big)=(x-2)^{2}\) at \(x=0,2\); integrate on \([0,2]\).
Vertical leg length: \(L(x)=4\cos\!\big(\tfrac{\pi x}{4}\big)-(x-2)^{2}\).
Area of isosceles right cross section: \(A(x)=\tfrac{1}{2}\,L(x)^{2}\).
Volume: \(\displaystyle V=\int_{0}^{2}\tfrac{1}{2}\Big(4\cos\!\big(\tfrac{\pi x}{4}\big)-(x-2)^{2}\Big)^{2}\,dx\).
Numerical value \(\approx 1.775\).
✅ Answer: (A) \(1.775\)

Calc-Ok Question


The base of a solid is a right triangle with legs \(6\) and \(16\). The hypotenuse is \(y=-\tfrac{3}{8}(x-16)\) on \(0\le x\le 16\). Each cross section perpendicular to the \(x\)-axis is a semicircle. What is the volume of the solid?
(A) \(75.398\)
(B) \(150.796\)
(C) \(301.593\)
(D) \(603.186\)
▶️ Answer/Explanation
For \(0\le x\le 16\), height \(y=6-\frac{3}{8}x\).
Semicircle radius \(r=\dfrac{y}{2}=3-\dfrac{3x}{16}\).
Area \(A(x)=\dfrac{\pi r^{2}}{2}=\dfrac{\pi}{2}\Big(3-\dfrac{3x}{16}\Big)^{2}\).
Volume \(\displaystyle V=\int_{0}^{16}A(x)\,dx=\frac{\pi}{2}\int_{0}^{16}\!\Big(3-\frac{3x}{16}\Big)^{2}dx =\frac{\pi}{2}\Big[9x-\frac{9}{16}x^{2}+\frac{9}{768}x^{3}\Big]_{0}^{16}=24\pi\approx 75.398.\)
✅ Answer: (A)
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