Find the area between $y = ( x - 3 ) e ^ { x }$ and the $x$-axis from $x = 3$ to $x = 6$.
A) $\mathrm { e } ^ { 6 } - \mathrm { e } ^ { 3 }$
B) $\mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$
C) $2 \mathrm { e } ^ { 6 }$
D) $2 \mathrm { e } ^ { 6 } + \mathrm { e } ^ { 3 }$

Question

Quizplus · Accepted Answer

To find the area between the curve and the x-axis, we need to integrate the function with respect to x from x = 3 to x = 6.

$$\begin{aligned} \int_{3}^{6}(x-3)e^{x}dx &= \left[e^{x}(x-4)\right]_{3}^{6} \ &= e^{6}(6-4)-e^{3}(3-4) \ &= 2e^{6}+e^{3} \end{aligned}$$

Therefore, the area between the curve and the x-axis is 2e^(6) + e^(3), which is answer choice D.

Find the Area Between y=(x−3)exy = ( x - 3 ) e ^ { x }y=(x−3)ex

Find the Area Between $y = ( x - 3 ) e ^ { x }$