Home / AP Calculus AB 8.6 Finding the Area Between Curves That Intersect at More Than Two Points – MCQs

AP Calculus AB 8.6 Finding the Area Between Curves That Intersect at More Than Two Points - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

No-Calc Question

What is the total area of the regions between the curves \(y=6x^{2}-18x\) and \(y=-6x\) from \(x=1\) to \(x=3\)?

(A) \(4\)
(B) \(12\)
(C) \(16\)
(D) \(20\)

▶️ Answer/Explanation

Intersections on \([1,3]\): solve \(6x^{2}-18x=-6x\Rightarrow 6x(x-2)=0\Rightarrow x=2\) (in interval).
On \([1,2]\), line above: area \(=\displaystyle \int_{1}^{2}\!\big[(-6x)-(6x^{2}-18x)\big]\,dx=\int_{1}^{2}\!(-6x^{2}+12x)\,dx\).
\(\displaystyle \int_{1}^{2}(-6x^{2}+12x)\,dx=\Big[-2x^{3}+6x^{2}\Big]_{1}^{2}=8-4=4\).
On \([2,3]\), parabola above: area \(=\displaystyle \int_{2}^{3}\!\big[(6x^{2}-18x)-(-6x)\big]\,dx=\int_{2}^{3}(6x^{2}-12x)\,dx\).
\(\displaystyle \int_{2}^{3}(6x^{2}-12x)\,dx=\Big[2x^{3}-6x^{2}\Big]_{2}^{3}=0-(-8)=8\).
Total area \(=4+8=12\).
✅ Answer: (B) \(12\)

Question
The region bounded by the x-axis and the part of the graph of \( y = \cos x \) between \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \) is separated into two regions by the line \( x = k \). If the area of the region for \( -\frac{\pi}{2} \leq x \leq k \) is three times the area of the region for \( k \leq x \leq \frac{\pi}{2} \), then \( k = \)
(A) \( \arcsin\left(\frac{1}{4}\right) \)
(B) \( \arcsin\left(\frac{1}{3}\right) \)
(C) \( \frac{\pi}{6} \)
(D) \( \frac{\pi}{4} \)
(E) \( \frac{\pi}{3} \)
▶️ Answer/Explanation
Solution
Area from \( x = -\frac{\pi}{2} \) to \( x = k \): \( \int_{-\frac{\pi}{2}}^{k} \cos x \, dx = \sin k -inf \sin\left(-\frac{\pi}{2}\right) = \sin k + 1 \).
Area from \( x = k \) to \( x = \frac{\pi}{2} \): \( \int_{k}^{\frac{\pi}{2}} \cos x \, dx = \sin\left(\frac{\pi}{2}\right) – \sin k = 1 – \sin k \).
Given: \( \sin k + 1 = 3 (1 – \sin k) \).
Solve: \( \sin k + 1 = 3 – 3 \sin k \), \( 4 \sin k = 2 \), \( \sin k = \frac{1}{2} \).
In \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), \( k = \frac{\pi}{6} \).
✅ Answer: C
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