Question

Sketch the region bounded by the curves y = ex, y = 0, x = 0, x = 5 and visually estimate the location of the centroid.

Answer 1

The region bounded by the curves y =
`e^x`, y = 0, x = 0, and x = 5 lies above the x-axis and to the right of the y-axis. The centroid is approximately located at
`\((\bar{x}, \bar{y}) \approx (3.45, 8.27)`.

Step-by-step explanation:

The given curves y =
`e^x` and y = 0 intersect at x = 0 and x = ln(5) (approximately 1.61). To estimate the centroid, divide the region into infinitesimally thin vertical strips along the x-axis. Using the centroid formula for continuous functions, calculate
`\(\bar{x}\)` and
`\(\bar{y}\)` as weighted averages of x and y respectively. For the area, integrate
`\(e^x\)` from 0 to ln(5) with respect to x, which equals
`\(e^(\ln(5)) - e^0` = 5 - 1 = 4.

To find
`\(\bar{x}\)`, integrate
`\(x \cdot e^x\)` over the same interval and divide by the area. This computation yields an approximate value of
`\(\bar{x} \approx 3.45\)`. For
`\(\bar{y}\)`, integrate
`\(0.5 \cdot e^(2x)\)` over the same interval and divide by the area, resulting in an approximate value of
`\(\bar{y} \approx 8.27\)`. These estimations suggest that the centroid lies closer to x = 3.45 and y = 8.27 within the bounded region.