Question

1. 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 )?

(a) ( int_{0}^{5}(5 - e^{y})dy )
(b) ( int_{1}^{5}(5 - e^{y})dy )
(c) ( int_{0}^{ln 5}(5 - e^{y})dy )
(d) ( int_{1}^{ln 5}(5 - e^{y})dy )

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