Answer each question appropriately.
-$$\text { Suppose that } \int _ { 0 } ^ { 3 } f ( x ) d x = 11 \text {. Find } \int _ { - 3 } ^ { 0 } f ( x ) d x \text {, if } f \text { is even. }$$
A) 33
B) -11
C) 11
D) -33

Question

Quizplus · Accepted Answer

If the function is even, then $\int_{-3}^{0} f(x) dx = \int_{0}^{3}f(-x) dx$. Since we know $\int_{0}^{3}f(x)dx = 11$, then $\int_{0}^{3}f(-x)dx = 11$. We can make the substitution $u = -x$, so that $dx = -du$ and the limits of integration change to $-3$ and $0$. Thus, $\int_{-3}^{0}f(x)dx = \int_{3}^{0}f(u)(-du) = \int_{0}^{3}f(u)du = \int_{0}^{3}f(x)dx = 11$.