Find the area.
-$\text { Find the area of the region between the curve } x = e ^ { 5 t } , y = \frac { 1 } { 5 } e ^ { - 4 t } \text { and the } x \text {-axis, } 0 \leq t \leq \ln 9 \text {. }$
A) 4
B) 9
C) 8
D) 5

Question

Quizplus · Accepted Answer

The area under the curve from $t = 0$ to $t = \ln 9$ can be found by integrating the function $y$ with respect to $t$, and since $y$ is given as a function of $t$, we need to express $dx$ in terms of $dt$ to perform the integration. Given $x = e^{5t}$, $dx = 5e^{5t}dt$. The area $A$ is then given by the integral of $y dx = \frac{1}{5}e^{-4t} \cdot 5e^{5t}dt = e^{t}dt$ from $t = 0$ to $t = \ln 9$. Thus, $A = \int_{0}^{\ln 9} e^{t} dt = [e^{t}]_{0}^{\ln 9} = e^{\ln 9} - e^{0} = 9 - 1 = 8$.

Find the Area A) 4 B) 9 C) 8 D) 5

Find the Area A) 4
B) 9
C) 8
D) 5