5. let ( r ) be the region between the graph of ( y=ln x ), the ( x )-axis, and the line ( x = 5 ). which of…

5. let ( r ) be the region between the graph of ( y=ln x ), the ( x )-axis, and the line ( x = 5 ). which of the following gives the area of region ( r )?\n(a) ( int_{0}^{5}(5 - e^{y})dy )\n(b) ( int_{1}^{5}(5 - e^{y})dy )\n(c) ( int_{0}^{ln 5}(5 - e^{y})dy )\n(d) ( int_{1}^{ln 5}(5 - e^{y})dy )

5. let ( r ) be the region between the graph of ( y=ln x ), the ( x )-axis, and the line ( x = 5 ). which of the following gives the area of region ( r )?\n(a) ( int_{0}^{5}(5 - e^{y})dy )\n(b) ( int_{1}^{5}(5 - e^{y})dy )\n(c) ( int_{0}^{ln 5}(5 - e^{y})dy )\n(d) ( int_{1}^{ln 5}(5 - e^{y})dy )

Answer

Explanation:

Step1: Find the intersection of (y = \ln x) and (y = 0)

Set (y=\ln x = 0), then (x = e^{0}=1). When (x = 5), (y=\ln5).

Step2: Use the formula for the area between two curves with respect to (y)

The area (A=\int_{a}^{b}(x_{right}-x_{left})dy). Here, (x_{right}=5) and (x_{left}=e^{y}), and the limits of integration for (y) are from (y = 0) to (y=\ln5). So the area (A=\int_{0}^{\ln5}(5 - e^{y})dy)

Answer:

C. (\int_{0}^{\ln5}(5 - e^{y})dy)