Graph the integrand and use areas to evaluate the integral.
-$$\int _ { - 7 } ^ { 7 } ( | x | + 4 ) d x$$
A) 77
B) 154
C) 105
D) 161

Question

Quizplus · Accepted Answer

The integrand, $|x|+4$, is a piecewise function and can be split into two cases: $x\geq 0$ and $x<0$.

For $x\geq 0$, $|x|=x$, so the integrand is $x+4$.

For $x<0$, $|x|=-x$, so the integrand is $-x+4$.

Therefore, we can evaluate the integral by finding the areas of the two regions separated by the $y$-axis and adding them together.

Region 1 (where $x\geq 0$): 
$$\int_{0}^{7}(x+4)dx = \frac{1}{2}(7^2)(7-0) + 4(7-0) = \frac{105}{2}$$

Region 2 (where $x<0$): 
$$\int_{-7}^{0}(-x+4)dx = \frac{1}{2}(0-(-7))^2(0-(-7)) + 4(0-(-7)) = \frac{49}{2}$$

Adding the areas of the two regions together, we get: 
$$\frac{105}{2} + \frac{49}{2} = 77$$

Therefore, the answer is (C) 105.

Graph the Integrand and Use Areas to Evaluate the Integral ∫−77(∣x∣+4)dx\int _ { - 7 } ^ { 7 } ( | x | + 4 ) d x∫−77​(∣x∣+4)dx