Question
The graph of y = ( fx) is shown in the figure above. If\( A_{1}\) and \(A_{2}\) are positive numbers that represent the areas of the shaded regions, then in terms of \( A_{1}\) and \(A_{2}\),
\(\int_{-4}^{4}f(x)dx-2\int_{-1}^{4}f(x)dx=\)
(A) \(A_{1}\) (B) \(A_{1}-A_{2}\) (C) \(2A_{1}-A_{2}\) (D) \(A_{1}+A_{2}\) (E) \(A _{1}+A_{2}\)
Answer/Explanation
Ans:C\D
Question
The area of the shaded region in the figure above is represented by which of the following integrals?
(A)\(\int_{a}^{c}\left ( |f(x)|-g(x) |\right )dx\)
(B)\(\int_{b}^{c} f(x)dx-\int_{a}^{c}g(x)dx\)
(C)\(\int_{a}^{c}(g(x)-f(x))dx\)
(D)\(\int_{a}^{c}(f(x)-g(x))dx\)
(E)\(\int_{a}^{b}(g(x)-f(x))dx+\int_{b}^{c}(f(x)-g(x))dx\)
Answer/Explanation
Ans:D
Question
Which of the following represents the area of the shaded region in the figure above?
(A) \(\int_{c}^{d}f(y)dy\) (B)\(\int_{a}^{b}(d-f(x))dx \) (C)\(f'(b)-f'(a) \) (D)\( (b-a)[f(b)-f(a)] \) (E)\((d-c)[f(b)-f(a)]\)
Answer/Explanation
Ans:B
Summing pieces of the form: (vertical). (small width), vertical = (d− f(x)), width = \(\Delta x\)
Question
The function f is continuous on the closed interval [ 0, 6] and has the values given in the table above. The trapezoidal approximation for\( \int_{0}^{6}f(x)\) dx found with 3 subintervals of equal length is 52. What is the value of k ?
(A) 2 (B) 6 (C) 7 (D) 10 (E) 14