10. Write and simplify the integral that gives the arc length of the curve y = x³ +500 for -1 ≤ x ≤ 2. Then use a left Riemann sum with n = 40 to approximate the length of the curve. Round your answer to four decimal places.
10. Write and simplify the integral that gives the arc length of the curve y = x³ +500 for -1 ≤ x ≤ 2. Then use a left Riemann sum with n = 40 to approximate the length of the curve. Round your answer to four decimal places.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Calculating the Arc Length of a Curve
**Problem Statement:**
1. **Write and simplify the integral that gives the arc length of the curve**
\[ y = x^3 + 500 \]
for
\[ -1 \leq x \leq 2. \]
2. **Then use a left Riemann sum with \( n = 40 \) to approximate the length of the curve. Round your answer to four decimal places.**
**Steps to Solve:**
1. **Set Up the Integral for Arc Length:**
The formula for the arc length \( L \) of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \) is:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
2. **Calculate the Derivative:**
Given \( y = x^3 + 500 \),
\[
\frac{dy}{dx} = 3x^2
\]
3. **Plug the Derivative into the Integral:**
The arc length integral becomes:
\[
L = \int_{-1}^{2} \sqrt{1 + (3x^2)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 9x^2} \, dx
\]
4. **Use a Left Riemann Sum:**
To approximate the integral using a left Riemann sum with \( n = 40 \):
- The interval \([-1, 2]\) is divided into 40 equal subintervals.
- The width of each subinterval is \(\Delta x = \frac{2 - (-1)}{40} = \frac{3}{40} = 0.075\).
The left Riemann sum is:
\[
L \approx \sum_{i=0}^{39} f\left(a + i\Delta x\right) \Delta x
\]
where \( f(x) = \sqrt{1 + 9x^2} \).
Plugging in the values:
\[
L \approx \sum_{i=0}^{39} \sqrt{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf5ca60f-6906-44bf-a492-477ae53fc791%2F06dd6bbf-9e04-4c33-941e-1e21eec0a7d8%2Fxj79x6j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Arc Length of a Curve
**Problem Statement:**
1. **Write and simplify the integral that gives the arc length of the curve**
\[ y = x^3 + 500 \]
for
\[ -1 \leq x \leq 2. \]
2. **Then use a left Riemann sum with \( n = 40 \) to approximate the length of the curve. Round your answer to four decimal places.**
**Steps to Solve:**
1. **Set Up the Integral for Arc Length:**
The formula for the arc length \( L \) of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \) is:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
2. **Calculate the Derivative:**
Given \( y = x^3 + 500 \),
\[
\frac{dy}{dx} = 3x^2
\]
3. **Plug the Derivative into the Integral:**
The arc length integral becomes:
\[
L = \int_{-1}^{2} \sqrt{1 + (3x^2)^2} \, dx = \int_{-1}^{2} \sqrt{1 + 9x^2} \, dx
\]
4. **Use a Left Riemann Sum:**
To approximate the integral using a left Riemann sum with \( n = 40 \):
- The interval \([-1, 2]\) is divided into 40 equal subintervals.
- The width of each subinterval is \(\Delta x = \frac{2 - (-1)}{40} = \frac{3}{40} = 0.075\).
The left Riemann sum is:
\[
L \approx \sum_{i=0}^{39} f\left(a + i\Delta x\right) \Delta x
\]
where \( f(x) = \sqrt{1 + 9x^2} \).
Plugging in the values:
\[
L \approx \sum_{i=0}^{39} \sqrt{
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