AP Calculus AB : 8.6 Finding the Area Between Curves That Intersect at More Than Two Points- Exam Style questions with Answer- MCQ

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 three times the area of the region for \(-\frac{\pi }{2}\leqslant x\leqslant\) 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)\(\frac{π}{4}\)         (E)  \(\frac{π}{3}\)

▶️Answer/Explanation

Ans:C

Question

Let R be the region bounded by the graphs of y=2x and \(y=4x-x^{2}\) What is the area of R ?

A \(\frac{2}{3}\)

B \(\frac{4}{3}\)

C \(\frac{16}{3}\)

D \(\frac{28}{3}\)

▶️Answer/Explanation

Ans:A

The graphs of   y=2x and \(y=4x-x^{2}\) intersect when x=0 and x=2. The graph of \(y=4x-x^{2}\) lies above the graph y=2x on the interval 0x2. (One way to see this is to sketch a graph of the parabola and the line, observing that the graph of \(y=4x-x^{2}\)   has a slope of 4 at x=0, while the graph of y=2x has a slope of 2) The area of the region bounded by the two graphs is therefore \(\int ^{2}_{0}(4x-x^{2}-2x)dx=\int _{0}^{2}(2-x^{2})dx=[x^{2}-\frac{x^{3}}{3}]|_{0}^{2}=4-\frac{8}{3}=\frac{4}{3}\)

Question

Calculate the approximate area of the shaded region in the figure by the trapezoidal rule, using divisions at   x = \(\frac{4}{3}\)  and x=\(\frac{5}{3}\)

(A)\(\frac{50}{27}\)                  (B)\(\frac{251}{108}\)                        (C)\(\frac{7}{3}\)                 (D)\(\frac{127}{54}\)               (E)\(\frac{77}{27}\)

▶️Answer/Explanation

Ans:D

Question

The area of the region in the first quadrant that is enclosed by the graphs of \(y=x^{3}+8\)  and y=x+8 is

(A) \(\frac{1}{4}\)                            (B) \(\frac{1}{2}\)                             (C) \(\frac{3}{4}\)                        (D) 1            (E) \(\frac{65}{4}\)

▶️Answer/Explanation

Ans:C

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