Graph the integrand and use areas to evaluate the integral.
-$$\int _ { - 7 } ^ { 7 } ( 7 - | x | ) d x$$
A) 98
B) $\frac { 49 } { 2 }$
C) 49
D) 147

Question

Quizplus · Accepted Answer

The integrand is symmetric about the y-axis, so we can split the integral into two parts, from 0 to 7 and from -7 to 0.

From 0 to 7, the integrand is given by $7-x$, so we can find the area under this part of the curve by finding the area of the triangle with base 7 and height 7, and then dividing by 2:

$	ext{Area} = \frac{1}{2}(7)(7) = \frac{49}{2}$

From -7 to 0, the integrand is given by $7+x$, so we can find the area under this part of the curve by finding the area of the triangle with base 7 and height 7, and then dividing by 2:

$	ext{Area} = \frac{1}{2}(7)(7) = \frac{49}{2}$

Therefore, the total area under the curve is $\frac{49}{2} + \frac{49}{2} = 49$, so the answer is (C).

Graph the Integrand and Use Areas to Evaluate the Integral ∫−77(7−∣x∣)dx\int _ { - 7 } ^ { 7 } ( 7 - | x | ) d x∫−77​(7−∣x∣)dx

Graph the Integrand and Use Areas to Evaluate the Integral $\int _ { - 7 } ^ { 7 } ( 7 - | x | ) d x$