Chapter 5.8, Problem 87E

To determine

(a)

Explain why the area of shaded region in Figure 5 is ${\displaystyle \underset{0}{\overset{\ln a}{\int }}{e}^{y}dy}$

To determine

(b)

To prove ${\displaystyle \underset{1}{\overset{a}{\int }}\ln xdx}=a\ln a-{\displaystyle \underset{0}{\overset{\ln a}{\int }}{e}^{y}dy}$

To determine

(c)

To prove ${\displaystyle \underset{1}{\overset{a}{\int }}\ln xdx}=a\ln a-a+1$

To determine

(d)

To find antiderivative of $y=\ln x$ with the use of result of (a)