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

question let the region r be the area enclosed the function f(x)=e^x, the horizontal line y = 17, and the y - axis. write an integral in terms of x and also an integral in terms of y that would represent the area of the region r. if necessary, round limit values to the nearest thousandth.

question let the region r be the area enclosed the function f(x)=e^x, the horizontal line y = 17, and the y - axis. write an integral in terms of x and also an integral in terms of y that would represent the area of the region r. if necessary, round limit values to the nearest thousandth.

Answer

Explanation:

Step1: Find intersection - point with (y = 17)

Set (y=e^{x}=17), then (x = \ln(17)\approx2.833).

Step2: Integral with respect to (x)

The area between the curves with respect to (x) is given by the integral of the upper - function minus the lower - function. The upper function is (y = 17) and the lower function is (y = e^{x}) from (x=0) to (x=\ln(17)). So the integral is (\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 with respect to (y)

The area between the curves with respect to (y) is given by integrating with respect to (y) from (y = 1) to (y = 17). The right - hand function is (x=\ln(y)) and the left - hand function is (x = 0). So the integral is (\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)