Graph the integrand and use areas to evaluate the integral.
-$$\int _ { - 3 } ^ { 3 } ( | x | + 10 ) d x$$
A) 39
B) 78
C) 69
D) 129

Question

Quizplus · Accepted Answer

The graph of $|x| + 10$ is a V-shape with its lowest point at (0, 10). From $-3$ to $0$, the line has a slope of 1, and from $0$ to $3$, it also has a slope of 1. The area under the curve from $-3$ to $3$ can be divided into two parts: two right triangles with bases of 3 units each and a height of 3 units (from the absolute value function), plus a rectangle under them with a length of 6 units and a height of 10 units. The area of each triangle is $\frac{1}{2} \times 3 \times 3 = 4.5$, so for both triangles, the area is $9$. The area of the rectangle is $6 \times 10 = 60$. Adding these areas together gives $9 + 60 = 69$.

Graph the Integrand and Use Areas to Evaluate the Integral ∫−33(∣x∣+10)dx\int _ { - 3 } ^ { 3 } ( | x | + 10 ) d x∫−33​(∣x∣+10)dx

Graph the Integrand and Use Areas to Evaluate the Integral $\int _ { - 3 } ^ { 3 } ( | x | + 10 ) d x$