Graph the integrand and use areas to evaluate the integral.
-$$\int _ { - 8 } ^ { 4 } ( - 2 x + 8 ) d x$$
A) 48
B) 192
C) 288
D) 144

Question

Quizplus · Accepted Answer

The integrand, $-2x+8$, is a linear function with $y$-intercept $8$ and slope $-2$.

Graphing this function, we get:

[asy]
import graph;
size(7cm);
Label f;
f.p=fontsize(6);
xaxis("-8",Arrows(4),Ticks(f, 1.0));
yaxis(Arrows(4),Ticks(f, 20.0));
real f(real x)
{
return -2x+8;
}
draw(graph(f,-8,4), linewidth(1));
label("$y=-2x+8$", (4,-16), red);
dot((4,0));
dot((-8,24));
fill((4,0)--(4,-8)--(-8,16)--(-8,24)--cycle, gray(0.7));
[/asy]

Note that the integrand is negative for $x>4$ and positive for $x<-8$, so the area above the $x$-axis and under the curve is split into two parts. We can find the area of each part separately and add them to get the total area.

For $-8\leq x\leq 4$, the integrand is nonnegative, so the area under the curve is given by: 
$$\int_{-8}^4 (-2x+8)\,dx = \left[-x^2+8xight]_{-8}^4 =\left[ -(4)^2+8(4) -(-8)^2+8(-8)ight]=144$$
(Note that we can check our answer by noting that the area of the trapezoid in the graph is $(1/2)(12)(8+24)=144$.)

Therefore, the correct answer is $\boxed{	ext{(D) }144}$.

Graph the Integrand and Use Areas to Evaluate the Integral ∫−84(−2x+8)dx\int _ { - 8 } ^ { 4 } ( - 2 x + 8 ) d x∫−84​(−2x+8)dx

Graph the Integrand and Use Areas to Evaluate the Integral $\int _ { - 8 } ^ { 4 } ( - 2 x + 8 ) d x$