Questions: Find the centroid of the region lying underneath the graph of the function (f(x)=e^-x) over the interval ([0,1]). [ xC M=(e-2) /(e-1) yC M= ]

Solution Steps

To find the centroid of the region under the curve \( f(x) = e^{-x} \) over the interval \([0, 1]\), we need to calculate the coordinates \( (x_{CM}, y_{CM}) \). The \( x \)-coordinate of the centroid \( x_{CM} \) is given, so we focus on finding \( y_{CM} \). The formula for \( y_{CM} \) is:

\[ y_{CM} = \frac{1}{A} \int_{a}^{b} \frac{1}{2} [f(x)]^2 \, dx \]

where \( A \) is the area under the curve from \( a \) to \( b \). First, calculate the area \( A \) using the integral of \( f(x) \) from 0 to 1. Then, use the formula for \( y_{CM} \) to find its value.

To find the area \( A \) under the curve \( f(x) = e^{-x} \) from \( x = 0 \) to \( x = 1 \), we compute the integral:

\[ A = \int_{0}^{1} e^{-x} \, dx = 1 - e^{-1} \]

Next, we calculate the integral needed for \( y_{CM} \):

\[ \int_{0}^{1} \frac{1}{2} [f(x)]^2 \, dx = \int_{0}^{1} \frac{1}{2} (e^{-x})^2 \, dx = \int_{0}^{1} \frac{1}{2} e^{-2x} \, dx = \frac{1}{4} - \frac{1}{4} e^{-2} \]

Now, we can find \( y_{CM} \) using the formula:

\[ y_{CM} = \frac{1}{A} \int_{0}^{1} \frac{1}{2} [f(x)]^2 \, dx = \frac{\frac{1}{4} - \frac{1}{4} e^{-2}}{1 - e^{-1}} \]

Evaluating this gives:

\[ y_{CM} \approx 0.34197 \]

Final Answer