Graph the integrand and use areas to evaluate the integral.
-$$\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x$$
A) 72
B) 18
C) 12
D) 36

Question

Quizplus · Accepted Answer

The integrand $2x+6$ is a linear function, which means its graph is a straight line. When we graph it, we get:

[asy]
import graph;
size(8cm);
Label f;
f.p=fontsize(6);
xaxis(-4,4,Ticks(f, 4.0));
yaxis(-1,17,Ticks(f, 4.0));
real f(real x)
{
return 2*x+6;
}
draw(graph(f,-3,3),linewidth(1));
fill((3,0)--(3,12)--(-3,6)--(-3,0)--cycle,gray);
[/asy]

The integral represents the signed area between the curve and the $x$-axis from $x=-3$ to $x=3$. We can split this area into a rectangle and a triangle, as shown in the graph. The rectangle has base $6$ (the distance between $-3$ and $3$) and height $6$ (the value of the integrand at $x=0$). Therefore, its area is $6\cdot6=36$. The triangle has base $6$ and height $6$ (the height of the rectangle minus the integrand at $x=3$), so its area is $(1/2)\cdot6\cdot6=18$. The total area is the sum of these two areas, which is $36+18=54$. However, we need to consider the signed area. Since the integrand is positive for all $x$ in our interval, the area is positive as well. Therefore, the value of the integral is equal to the absolute value of the area, which is $|54|=54$.

Therefore, the correct choice is D) 36.

Graph the Integrand and Use Areas to Evaluate the Integral ∫−33(2x+6)dx\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x∫−33​(2x+6)dx

Graph the Integrand and Use Areas to Evaluate the Integral $\int _ { - 3 } ^ { 3 } ( 2 x + 6 ) d x$