Question

Consider the function $f\left(x\right)=\ln x$.

(a) Find the signed area between the graph of $f\left(x\right)$and the $x$-axis on $\left[\frac{1}{2},4\right]$shown in the figure next at the left.

(b) Find the area between the graph of $f\left(x\right)$and the graph of $g\left(x\right)=1$on $\left[\frac{1}{2},4\right]$shown in the figure next at the right.

Short answer

Part (a). The solution is $-\frac{7}{2}+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)$.

Part (b). The solution is$-7+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)$.

Step-by-step solution

Part (a) Step 1. Given information.

The given function is$f\left(x\right)=\ln x$.

Part (a) Step 2. Solve the definite integral ∫124lnx dx.

Apply integration by parts and solve.

$\begin{array}{rcl}{\int }_{\frac{1}{2}}^4\ln xdx& =& {\left[\ln x\int dx-\int \left\{\frac{d}{dx}\left(\ln x\right)\int dx\right\}dx\right]}_{\frac{1}{2}}^4\\ & =& {\left[\ln x·x-\int \frac{1}{x}·xdx\right]}_{\frac{1}{2}}^4\\ & =& {\left[\ln x·x-\int dx\right]}_{\frac{1}{2}}^4\\ & =& {\left[\ln x·x-x\right]}_{\frac{1}{2}}^4\\ & =& \left[4\ln 4-4\right]-\left[\frac{1}{2}\ln \frac{1}{2}-\frac{1}{2}\right]\\ & =& -\frac{7}{2}+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)\end{array}$

Part (b) Step 1. Given information.

The given function is$f\left(x\right)=\ln x$and$g\left(x\right)=1$.

Part (b) Step 2. Find the area between the graph of f(x)=lnx and g(x)=1 on 12,4.

The area is calculated below.

$\begin{array}{rcl}{\int }_{\frac{1}{2}}^4\left(f\left(x\right)-g\left(x\right)\right)dx& =& {\int }_{\frac{1}{2}}^4\left[\ln x-1\right]dx\\ & =& {\int }_{\frac{1}{2}}^4\ln xdx-{\int }_{\frac{1}{2}}^{4}dx\\ & =& {\left[x\ln x-x\right]}_{\frac{1}{2}}^4-{\left[x\right]}_{\frac{1}{2}}^4\\ & =& \frac{-7}{2}+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)-\left[4-\frac{1}{2}\right]\\ & =& \frac{-7}{2}+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)-\frac{7}{2}\\ & =& -\left\{\frac{7+7}{2}\right\}+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)\\ & =& -7+\frac{1}{2}\ln \left(2\right)+4\ln \left(4\right)\end{array}$