a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. y = x², -1 ≤ x ≤ 2
a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. y = x², -1 ≤ x ≤ 2
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 44E
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![### Length of a Curve
To determine the length of a curve defined by a function, follow these steps:
#### a. Set up an integral for the length of the curve
Given the function \( y = x^2 \) over the interval \(-1 \leq x \leq 2\), the formula for the length \( L \) of a curve from \( x = a \) to \( x = b \) is given by:
\[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx} \right)^2} \, dx \]
First, find the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2x \]
Then, substitute into the formula:
\[ L = \int_{-1}^2 \sqrt{1 + (2x)^2} \, dx \]
#### b. Graph the curve to see what it looks like
Graph the function \( y = x^2 \) on the interval \(-1 \leq x \leq 2\). This quadratic function will produce a parabolic curve opening upward. The key points to plot can include \( x = -1 \), \( x = 0 \), \( x = 1 \), and \( x = 2 \):
- At \( x = -1 \), \( y = (-1)^2 = 1 \)
- At \( x = 0 \), \( y = 0^2 = 0 \)
- At \( x = 1 \), \( y = 1^2 = 1 \)
- At \( x = 2 \), \( y = 2^2 = 4 \)
#### c. Use your grapher’s or computer’s integral evaluator to find the curve’s length numerically
To evaluate this integral numerically, use a graphing calculator or computer software capable of performing definite integrals. Input the integral:
\[ \int_{-1}^2 \sqrt{1 + 4x^2} \, dx \]
The output will provide the numerical value for the length of the curve between \( x = -1 \) and \( x = 2 \).
By following these steps, you can determine the length of the curve defined by \( y = x^2 \) over the given interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a617c73-16c8-405c-b9c4-72dd4741d892%2Fe1e2746e-25ec-4463-9361-c2b9388d72c1%2Fco1sdcd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Length of a Curve
To determine the length of a curve defined by a function, follow these steps:
#### a. Set up an integral for the length of the curve
Given the function \( y = x^2 \) over the interval \(-1 \leq x \leq 2\), the formula for the length \( L \) of a curve from \( x = a \) to \( x = b \) is given by:
\[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx} \right)^2} \, dx \]
First, find the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2x \]
Then, substitute into the formula:
\[ L = \int_{-1}^2 \sqrt{1 + (2x)^2} \, dx \]
#### b. Graph the curve to see what it looks like
Graph the function \( y = x^2 \) on the interval \(-1 \leq x \leq 2\). This quadratic function will produce a parabolic curve opening upward. The key points to plot can include \( x = -1 \), \( x = 0 \), \( x = 1 \), and \( x = 2 \):
- At \( x = -1 \), \( y = (-1)^2 = 1 \)
- At \( x = 0 \), \( y = 0^2 = 0 \)
- At \( x = 1 \), \( y = 1^2 = 1 \)
- At \( x = 2 \), \( y = 2^2 = 4 \)
#### c. Use your grapher’s or computer’s integral evaluator to find the curve’s length numerically
To evaluate this integral numerically, use a graphing calculator or computer software capable of performing definite integrals. Input the integral:
\[ \int_{-1}^2 \sqrt{1 + 4x^2} \, dx \]
The output will provide the numerical value for the length of the curve between \( x = -1 \) and \( x = 2 \).
By following these steps, you can determine the length of the curve defined by \( y = x^2 \) over the given interval.
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