Chapter 6: Q. 39 (page 499)

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.
$f (x) = \frac{1}{3} x^{3 / 2} - x^{1 / 2}$ , $[a, b] = [0, 1]$

Short Answer

Expert verified

The arc length is $\frac{4}{3}$ .

Step by step solution

Step 1. Given information.

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

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 $\int_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x$ .

Step 3. Find the arc length.

$\begin{array}{rcl} Arc length & = & \int_{0}^{1} \sqrt{1 + {(\frac{d}{d x} (\frac{1}{3} x^{3 / 2} - x^{1 / 2}))}^{2}} d x \\ = & \int_{0}^{1} \sqrt{1 + {(\frac{1}{3} (\frac{3}{2}) x^{1 / 2} - (\frac{1}{2}) x^{- 1 / 2})}^{2}} d x \\ = & \int_{0}^{1} \sqrt{1 + {(\frac{1}{2} x^{1 / 2} - (\frac{1}{2}) x^{- 1 / 2})}^{2}} d x \\ = & \int_{0}^{1} \sqrt{1 + \frac{1}{4} {(\frac{x - 1}{\sqrt{x}})}^{2}} d x \\ = & \int_{0}^{1} \sqrt{1 + \frac{1}{4} (\frac{{(x - 1)}^{2}}{x})} d x \\ = & \int_{0}^{1} \sqrt{\frac{4 x + {(x - 1)}^{2}}{4 x}} d x \\ = & \frac{1}{2} \int_{0}^{1} \frac{x + 1}{\sqrt{x}} d x \\ = & \frac{1}{2} (\int_{0}^{1} \sqrt{x} d x + \int_{0}^{1} \frac{1}{\sqrt{x}} d x) \\ = & \frac{1}{2} ({[\frac{x^{3 / 2}}{3 / 2}]}_{0}^{1} + {[\frac{x^{1 / 2}}{1 / 2}]}_{0}^{1}) \\ = & \frac{1}{2} ([\frac{1^{3 / 2}}{3 / 2} - \frac{0^{3 / 2}}{3 / 2}] + [\frac{1^{1 / 2}}{1 / 2} - \frac{0^{1 / 2}}{1 / 2}]) \\ = & \frac{1}{2} (\frac{2}{3} + 2) \\ = & \frac{4}{3} \end{array}$

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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.
$f (x) = \frac{1}{3} x^{3 / 2} - x^{1 / 2}$ , $[a, b] = [0, 1]$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

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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(x)=13x3/2-x1/2,a,b=0,1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find the arc length.

One App. One Place for Learning.

Most popular questions from this chapter

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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.
$f (x) = \frac{1}{3} x^{3 / 2} - x^{1 / 2}$ , $[a, b] = [0, 1]$