Question

use geometry (not riemann sums) to evaluate the definite integral. sketch the graph of the integrand, show the region in question, and interpret your result. $int_{-4}^{3}(-2|x|)dx$. choose the correct graph below.

Explanation:

Step1: Split integral at x=0

$\int_{-4}^{3} -2|x|dx = \int_{-4}^{0} -2|x|dx + \int_{0}^{3} -2|x|dx$

Step2: Evaluate first integral (x<0)

For $x<0$, $|x|=-x$, so $-2|x|=2x$. Area of triangle: $\frac{1}{2} \times 4 \times 8=16$. Integral: $-16$

Step3: Evaluate second integral (x≥0)

For $x≥0$, $|x|=x$, so $-2|x|=-2x$. Area of triangle: $\frac{1}{2} \times 3 \times 6=9$. Integral: $-9$

Step4: Sum results

$-16 + (-9) = -25$

Answer:

-25