let the region r be the area enclosed the function f(x)=e^x, the horizontal line y = 17, and the y - axis…

Explanation:

Step1: Find intersection point of $y = e^x$ and $y = 17$

Set $e^x=17$, then $x=\ln(17)\approx2.833$.

Step2: Integral in terms of $x$

The upper - curve is $y = 17$ and the lower - curve is $y = e^x$ from $x = 0$ to $x=\ln(17)$. The area $A_x=\int_{0}^{\ln(17)}(17 - e^x)dx$.

Step3: Express $x$ in terms of $y$

Since $y = e^x$, then $x=\ln(y)$.

Step4: Integral in terms of $y$

The left - hand side is $x = 0$ and the right - hand side is $x=\ln(y)$. The limits of integration for $y$ are from $y = 1$ (when $x = 0$, $y=e^0 = 1$) to $y = 17$. The area $A_y=\int_{1}^{17}\ln(y)dy$.

Answer:

Integral in terms of $x$: $\int_{0}^{\ln(17)}(17 - e^x)dx$; Integral in terms of $y$: $\int_{1}^{17}\ln(y)dy$