Q. 79 - Calculus

Question

Consider the function $f\left(x\right)=\frac{x}{x^2+1}$.

(a) Find the signed area between the graph of f(x) and the x-axis on [−1, 3] shown next at the left.

(b) Find the area between the graph of f(x) and the graph of $g\left(x\right)=\frac{1}{4}x$ on [−1, 3] shown next at the right.

Step-by-Step Solution

Answer

(a) The area between the graph of f(x) and the x-axis on [−1, 3] is ${\int }_{-1}^{3}f\left(x\right)dx=\frac{1}{2}\left[\ln 10-\ln 2\right]$.

(b) The area between the graph of f(x) and the graph of $g\left(x\right)=\frac{1}{4}x$ on [−1, 3] is $\mathrm{Area}=\frac{1}{2}\left[\ln 10-\ln 2\right]-1$.

Step 1. Given Information

Consider the function $f\left(x\right)=\frac{x}{x^2+1}$.

(a) Find the signed area between the graph of f(x) and the x-axis on [−1, 3] shown next at the left.

(b) Find the area between the graph of f(x) and the graph of $g\left(x\right)=\frac{1}{4}x$ on [−1, 3] shown next at the right.

Part(a) Step 1. Now finding the area between the graph of f(x) and the x -axis on [−1, 3].

Using the substitution method. 

Let

$$\begin{gathered} u=x^2+1 \\ \frac{du}{dx}=2x \\ du=2xdx \\ \frac{1}{2}du=xdx \end{gathered}$$

Part (a) Step 2. We will now write the limits of integration in terms of the new variable u .

When $x=-1$, we have

$$\begin{gathered} u=x^2+1 \\ u={(-1)}^2+1 \\ u=1+1 \\ u=2 \end{gathered}$$

When $x=3$, we have

$$\begin{gathered} u=x^2+1 \\ u={\left(3\right)}^2+1 \\ u=9+1 \\ u=10 \end{gathered}$$

Part (a) Step 3. Using the information in equations, we can change variables completely:

$$\begin{gathered} {\int }_{-1}^{3}f\left(x\right)dx={\int }_{-1}^3\frac{x}{x^2+1}dx \\ {\int }_{-1}^{3}f\left(x\right)dx=\frac{1}{2}{\int }_2^{10}\frac{1}{u}du \\ {\int }_{-1}^{3}f\left(x\right)dx=\frac{1}{2}{\left[\ln u\right]}_2^{10} \\ {\int }_{-1}^{3}f\left(x\right)dx=\frac{1}{2}\left[\ln 10-\ln 2\right] \end{gathered}$$

Part(b) Step 1. Now finding the area between the graph of f(x) and the graph of g ( x ) = 1 4 x on [−1, 3] shown next at the right.

$$\begin{gathered} \mathrm{Area}={\int }_{-1}^3\left[\mathrm{f}\left(\mathrm{x}\right)-\mathrm{g}\left(\mathrm{x}\right)\right]\mathrm{dx} \\ \mathrm{Area}={\int }_{-1}^3\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}-{\int }_{-1}^3\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Part (b) Step 2. Firstly finding the value of ∫ - 1 3 g ( x ) d x

$$\begin{gathered} {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{4}{\int }_{-1}^{3}xdx \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{4}{\left[\frac{x^{1+1}}{1+1}\right]}_{-1}^3 \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{4}{\left[\frac{x^2}{2}\right]}_{-1}^3 \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{4}·\frac{1}{2}{\left[x^2\right]}_{-1}^3 \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{8}\left[{\left(3\right)}^2-{(-1)}^2\right] \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{8}\left[9-1\right] \\ {\int }_{-1}^{3}g\left(x\right)dx=\frac{1}{8}\left[8\right] \\ {\int }_{-1}^{3}g\left(x\right)dx=1 \end{gathered}$$

Part (b) Step 3. Putting the value of ∫ - 1 3 f ( x ) d x   and   ∫ - 1 3 g ( x ) d x in src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="48" width="216" wrs:baseline="29"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Area</mi><mo>=</mo><msubsup><mo>&#x222B;</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>3</mn></msubsup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mo>-</mo><msubsup><mo>&#x222B;</mo><mrow><mo>-</mo><mn>1</mn></mrow><mn>3</mn></msubsup><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></math>--><defs><style 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font-family="Arial" font-size="16" text-anchor="middle" x="17.5" y="29">Area</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="16" text-anchor="middle" x="42.5" y="29">=</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="32" text-anchor="middle" x="59.5" y="33">&#x222B;</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="12" text-anchor="middle" x="67.5" y="44">&#x2212;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="75.5" y="44">1</text><text font-family="Arial" font-size="12" text-anchor="middle" x="72.5" y="11">3</text><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="82.5" y="29">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="91.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="97.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="104.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="111.5" y="29">d</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="120.5" y="29">x</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="16" text-anchor="middle" x="132.5" y="29">&#x2212;</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="32" text-anchor="middle" x="148.5" y="33">&#x222B;</text><text font-family="math1951e8e33c6f57e36fd3785f029" font-size="12" text-anchor="middle" x="156.5" y="44">&#x2212;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="164.5" y="44">1</text><text font-family="Arial" font-size="12" text-anchor="middle" x="161.5" y="11">3</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="172.5" y="29">g</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="181.5" y="29">(</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="187.5" y="29">x</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="194.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="201.5" y="29">d</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="210.5" y="29">x</text></svg>" role="math" localid="1649090732221" Area = ∫ - 1 3 f ( x ) d x - ∫ - 1 3 g ( x ) d x

From the part (a) we know the value of 

${\int }_{-1}^{3}f\left(x\right)dx=\left[\ln 10-\ln 2\right]$

$$\begin{gathered} \mathrm{Area}={\int }_{-1}^{3}f\left(x\right)dx-{\int }_{-1}^{3}g\left(x\right)dx \\ \mathrm{Area}=\frac{1}{2}\left[\ln 10-\ln 2\right]-1 \end{gathered}$$