Question

let r be the shaded region bounded by $f(x) = 5e^{-\frac{x}{5}} + 3$, $x = 5$, and the $x$ and $y$ axes. find the perimeter of the shaded region. use a calculator and round your answer to 3 decimal places.

Explanation:

Step1: Find the \( y \)-intercept

When \( x = 0 \), \( f(0)=5e^{0}+3=5\times1 + 3=8 \).

Step2: Find the value of \( f(5) \)

When \( x = 5 \), \( f(5)=5e^{-1}+3=\frac{5}{e}+3\approx\frac{5}{2.718}+3\approx1.84+3 = 4.84 \).

Step3: Calculate the length of the curve

The formula for the arc - length of a function \( y = f(x) \) from \( x=a \) to \( x = b \) is \( L=\int_{a}^{b}\sqrt{1+(f^{\prime}(x))^{2}}dx \).
First, find \( f^{\prime}(x) \):
If \( f(x)=5e^{-\frac{x}{5}}+3 \), then \( f^{\prime}(x)=5\times(-\frac{1}{5})e^{-\frac{x}{5}}=-e^{-\frac{x}{5}} \).
So, \( (f^{\prime}(x))^{2}=e^{-\frac{2x}{5}} \), and \( 1+(f^{\prime}(x))^{2}=1 + e^{-\frac{2x}{5}} \).
The arc - length \( L=\int_{0}^{5}\sqrt{1 + e^{-\frac{2x}{5}}}dx \). Using a calculator, \( L\approx6.103 \).

Step4: Calculate the sum of the other sides

The length of the \( x \)-axis side is \( x = 5 \), and the length of the \( y \)-axis side is \( y=f(0) = 8 \). The length of the vertical side at \( x = 5 \) is \( f(5)\approx4.84 \).
The perimeter \( P=8 + 5+4.84+L \).
Substitute \( L\approx6.103 \) into the formula: \( P=8 + 5+4.84+6.103 \).

Answer:

\( 23.943 \)