Let be a continuous function that satisfies for all real and on

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

f

be a continuous function that satisfies

f(-x)=f(x)

for all real

x

and

f(x)\ge 0

[-a,a]

. Which statement best justifies why the area enclosed by the graph of

y=f(x)

and the

x

-axis on

[-a,a]

equals

2\displaystyle\int_{0}^{a} f(x)\,dx

? (a) Because

f

is continuous, its integral over any symmetric interval equals twice the average value of

f

[0,a]

, so the total area is

2\displaystyle\int_{0}^{a} f(x)\,dx

.},{ (b) Because

f

is even, the region on

[-a,0]

is a mirror image of the region on

[0,a]

with identical area. (c) Because

f

is even, every antiderivative of

f

is also even; therefore the definite integral doubles when the limits are symmetric. (d) Because

f

is odd, the integrals on

[-a,0]

and

[0,a]

cancel, so you must take twice the positive part on

[0,a]

to obtain the total area.