Find the Area of the region

Question

Sketch the region bounded by the curve 6xe^−x, the x-axis, and x = 3, and find the area of the region. Round your answer to four decimal places.

Area =



Answer 1

From 6xe^-x differentiate 6x downward and integrate e^-x downward as shown in the chart:

6x------------------e^-x

6------------------(-e^-x)

0-------------------(e^-x)

Write the chart above on paper and draw a diagonal arrow through 6x and (-e^-x). Mark that diagonal with a + sign.

Draw a second diagonal arrow through 6 and (e^-x). Mark this second diagonal with a - sign.

Now construct the integral as +[6x times (-e^-x)] − [6 times (e^-x)] or -6xe^-x − 6e^-x.

Then compute the area sought as ∫_{(from x=0 to x=3)}6xe^-xdx or [-6xe^-x − 6e^-x|_{(from x=0 to x=3)}].

Write -6(3)e^-3 − 6e^-3 − -6(0)e^-(0) − - 6e^-0 or -24e^-3 + 6.

6 − 24e^-3 calculates to 4.805110359 or 4.8051 square units.