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Answers without the blur. Sign up and see all textbooks for free! Q. 36

Expert-verified Found in: Page 499 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Find the exact value of the arc length of each function f(x) on [a, b] by writing the arc length as a definite integral and then solving that integral. $f\left(x\right)=2{\left(1-x\right)}^{3/2}+3$, $\left[a,b\right]=\left[-2,0\right]$

The arc length is $\begin{array}{rcl}& & \frac{2}{27}{\left(28\right)}^{3/2}-\frac{2}{27}{\left(10\right)}^{3/2}\end{array}$.

See the step by step solution

## Step 1. Given information.

Consider the given function $f\left(x\right)=2{\left(1-x\right)}^{3/2}+3$, $\left[a,b\right]=\left[-2,0\right]$.

## Step 2. Use arc length formula.

The formula for a function to find the arc length from $x=a$ to $x=b$ is given by ${\int }_{a}^{b}\sqrt{1+\left({f}^{\text{'}}{\left(x\right)\right)}^{2}}dx$.

## Step 3. Find the arc length.

$\begin{array}{rcl}\mathrm{Arc}\mathrm{length}& =& {\int }_{-2}^{0}\sqrt{1+{\left(\frac{\mathrm{d}}{\mathrm{dx}}\left(2{\left(1-\mathrm{x}\right)}^{3/2}+3\right)\right)}^{2}}\mathrm{dx}\\ & =& {\int }_{-2}^{0}\sqrt{1+{\left(\left(2{\left(\frac{3}{2}\right)\left(1-\mathrm{x}\right)}^{1/2}\left(-1\right)+0\right)\right)}^{2}}\mathrm{dx}\\ & =& {\int }_{-2}^{0}\sqrt{1+{\left(-3{\left(1-\mathrm{x}\right)}^{1/2}\right)\right)}^{2}}\mathrm{dx}\\ & =& {\int }_{-2}^{0}\sqrt{1+9\left(1-\mathrm{x}\right)}\mathrm{dx}\\ & =& {\int }_{-2}^{0}\sqrt{10-9\mathrm{x}}\mathrm{dx}\end{array}$

## Step 4. Simply obtained integral by simple substitution.

Substitute $10-9x=u$, $dx=-\frac{1}{9}du$ into $\begin{array}{rcl}& & {\int }_{-2}^{0}\sqrt{10-9\mathrm{x}}\mathrm{dx}\end{array}$.

$\begin{array}{rcl}{\int }_{28}^{10}-\frac{\sqrt{u}}{9}du& =& \left(-\frac{1}{9}\right){\int }_{28}^{10}\sqrt{u}du\\ & =& -\left(-\frac{1}{9}\right){\int }_{10}^{28}\sqrt{u}du\\ & =& \left(\frac{1}{9}\right){\int }_{10}^{28}\left(\frac{2}{3}{u}^{3/2}\right)\\ & =& \left(\frac{1}{9}\right){\left[\frac{2}{3}{u}^{3/2}\right]}_{10}^{28}\\ & =& \frac{2}{27}{\left(28\right)}^{3/2}-\frac{2}{27}{\left(10\right)}^{3/2}\end{array}$ ### Want to see more solutions like these? 