3. Evaluate I° + dz by taking a regular partition of an appropriate interval, using the right endpoint of each subinterval and taking a limit
3. Evaluate I° + dz by taking a regular partition of an appropriate interval, using the right endpoint of each subinterval and taking a limit
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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![**Problem 3: Evaluate the Integral**
Evaluate the integral \(\int_{2}^{4} (x^3 + x^2) \, dx\) by taking a regular partition of an appropriate interval, using the right endpoint of each subinterval, and taking a limit.
**Explanation:**
To solve this problem, you divide the interval [2, 4] into \(n\) equal subintervals. The width of each subinterval \(\Delta x\) is given by:
\[
\Delta x = \frac{4 - 2}{n} = \frac{2}{n}
\]
The right endpoint of each subinterval is:
\[
x_i = 2 + i \frac{2}{n} \quad \text{for} \quad i = 1, 2, \ldots, n
\]
The Riemann sum for this integral using the right endpoints is:
\[
\sum_{i=1}^{n} \left(\left(x_i^3 + x_i^2\right) \Delta x\right)
\]
Substitute \(x_i\) and \(\Delta x\):
\[
= \sum_{i=1}^{n} \left(\left(2 + i \frac{2}{n}\right)^3 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}
\]
Take the limit as \(n \to \infty\):
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left(\left(2 + i \frac{2}{n}\right)^3 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}
\]
This expression evaluates to the definite integral \(\int_{2}^{4} (x^3 + x^2) \, dx\).
To solve this using the Fundamental Theorem of Calculus:
Find the antiderivative of \(x^3 + x^2\):
\[
F(x) = \frac{x^4}{4} + \frac{x^3}{3}
\]
Evaluate \(F(x)\) from 2 to 4:
\[
F(4) - F(2) = \left(\frac{4^4}{4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f250653-9142-47c1-b115-0b4540b00021%2F2cfa0cd0-73c5-466c-8da9-2a36ff329ded%2Fju6h9ni_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Evaluate the Integral**
Evaluate the integral \(\int_{2}^{4} (x^3 + x^2) \, dx\) by taking a regular partition of an appropriate interval, using the right endpoint of each subinterval, and taking a limit.
**Explanation:**
To solve this problem, you divide the interval [2, 4] into \(n\) equal subintervals. The width of each subinterval \(\Delta x\) is given by:
\[
\Delta x = \frac{4 - 2}{n} = \frac{2}{n}
\]
The right endpoint of each subinterval is:
\[
x_i = 2 + i \frac{2}{n} \quad \text{for} \quad i = 1, 2, \ldots, n
\]
The Riemann sum for this integral using the right endpoints is:
\[
\sum_{i=1}^{n} \left(\left(x_i^3 + x_i^2\right) \Delta x\right)
\]
Substitute \(x_i\) and \(\Delta x\):
\[
= \sum_{i=1}^{n} \left(\left(2 + i \frac{2}{n}\right)^3 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}
\]
Take the limit as \(n \to \infty\):
\[
\lim_{n \to \infty} \sum_{i=1}^{n} \left(\left(2 + i \frac{2}{n}\right)^3 + \left(2 + i \frac{2}{n}\right)^2\right) \frac{2}{n}
\]
This expression evaluates to the definite integral \(\int_{2}^{4} (x^3 + x^2) \, dx\).
To solve this using the Fundamental Theorem of Calculus:
Find the antiderivative of \(x^3 + x^2\):
\[
F(x) = \frac{x^4}{4} + \frac{x^3}{3}
\]
Evaluate \(F(x)\) from 2 to 4:
\[
F(4) - F(2) = \left(\frac{4^4}{4
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