Q. 53 You may have noticed that even v... [FREE SOLUTION]

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Chapter 6: Q. 53 (page 499)

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. Use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b].
$f (x) = x^{3}$ , $[a, b] = [- 1, 1]$

Short Answer

Expert verified

The approximate value is $3.0957$ .

Step by step solution

Step 1. Given information.

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

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

Step 3. Find definite integral for the given function.

Substitute corresponding values into the arc length formula.

$\begin{array}{rcl} {鈭�}_{a}^{b} \sqrt{1 + {(f^{'} (x))}^{2}} d x & = & {鈭�}_{- 1}^{1} \sqrt{1 + {(\frac{d}{d x} (x^{3}))}^{2}} d x \\ = & {鈭�}_{- 1}^{1} \sqrt{1 + {(3 x^{2})}^{2}} d x \\ = & {鈭�}_{- 1}^{1} \sqrt{1 + 9 x^{4}} d x \end{array}$

Step 4. Use graphing calculator.

Find the approximate value of definite integral $\begin{array}{rcl} {鈭�}_{- 1}^{1} \sqrt{1 + 9 x^{4}} d x \end{array}$ with the help of graphing calculator.

$\begin{array}{rcl} 鈭� \end{array} \begin{array}{rcl} _{- 1}^{1} \end{array} \begin{array}{rcl} \sqrt{1 + 9 x^{4}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} 鈮� \end{array} 3.0957$

The area under the definite integral between the interval $[- 1, 1]$ in the graphing calculator is represented as follows.

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Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find definite integral for the given function.

Step 4. Use graphing calculator.

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

You may have noticed that even very simple functions give rise to arc length integrals that we have no idea how to compute. Use a graphing calculator to approximate a definite integral that represents the arc length of the given function f(x) on the interval [a, b].f(x)=x3,a,b=-1,1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use arc length formula.

Step 3. Find definite integral for the given function.

Step 4. Use graphing calculator.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.