007B Sample Midterm 2, Problem 4 Detailed Solution

Find the area of the region bounded by  $y=\ln x,~y=0,~x=1,$  and  $x=e.$

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Background Information:
1. You can find the intersection points of two functions, say  $f(x),g(x),$
by setting  $f(x)=g(x)$  and solving for  $x.$
2. The area between two functions,  $f(x)$  and  $g(x),$  is given by  $\int _{a}^{b}f(x)-g(x)~dx$
for  $a\leq x\leq b,$  where  $f(x)$  is the upper function and  $g(x)$  is the lower function.
3. Integration by parts tells us that
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int u~dv=uv-\int v~du.}

Solution:

Step 1:
We start by finding the intersection points of the functions  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\ln x}   and  $y=0.$
So, we consider the equation  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0=\ln x.}
The only solution to this equation is  $x=1.$
Also, for  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1<x<e,}   we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0<\ln(x).}

Step 2:
The area bounded by these functions is given by
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int _{1}^{e}\ln x~dx.}
Now, we need to use integration by parts.
Let  $u=\ln x$  and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle dv=dx.}
Then,  $du={\frac {1}{x}}~dx$  and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=x.}
Therefore, we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\int_1^{e} \ln x~dx} & = & \displaystyle{x\ln(x)\bigg|_1^e-\int_1^e dx}\\ &&\\ & = & \displaystyle{x\ln (x)-x\bigg|_1^e}\\ &&\\ & = & \displaystyle{e\ln (e)-e - (\ln(1)-1)}\\ &&\\ & = & \displaystyle{e-e-(0-1)}\\ &&\\ & = & \displaystyle{1.} \end{array}}

Final Answer:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}

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