**Topic: Calculus - Midpoint Rule****Problem 8.8.4**If the Midpoint Rule is used on the interval $[-3, 27]$ with $n = 3$ subintervals, at what x-coordinates is the integrand evaluated?**Instructions:** (Simplify your answer. Use a comma to separate answers as needed.)**Explanation:** To find the x-coordinates for evaluating the integrand using the Midpoint Rule with $n = 3$ subintervals across the interval $[-3, 27]$:1. **Determine the width of each subinterval:** The length of the interval is $27 - (-3) = 30$. With 3 subintervals, the width ($\Delta x$) of each subinterval is $30/3 = 10$.2. **Find the midpoints of each subinterval:** - The first subinterval is $[-3, 7]$. The midpoint is $(-3+7)/2 = 2$. - The second subinterval is $[7, 17]$. The midpoint is $(7+17)/2 = 12$. - The third subinterval is $[17, 27]$. The midpoint is $(17+27)/2 = 22$.**Solution:** The x-coordinates at which the integrand is evaluated are $2, 12, 22$.

given Interval [-3,27] and n = 3, the number of sub intervals. To find at what x-coordinate at…

Answered: If the Midpoint Rule is used on the…

If the Midpoint Rule is used on the interval [- 3,27] with n = 3 subintervals, at what x-coordinates is the integrand evaluated?

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differential equations. The most important aspect of this theory is error analysis.

Question

**Topic: Calculus - Midpoint Rule**

**Problem 8.8.4**

If the Midpoint Rule is used on the interval $[-3, 27]$ with $n = 3$ subintervals, at what x-coordinates is the integrand evaluated?

**Instructions:**
(Simplify your answer. Use a comma to separate answers as needed.)

**Explanation:**
To find the x-coordinates for evaluating the integrand using the Midpoint Rule with $n = 3$ subintervals across the interval $[-3, 27]$:

1. **Determine the width of each subinterval:**
The length of the interval is $27 - (-3) = 30$.
With 3 subintervals, the width ($\Delta x$) of each subinterval is $30/3 = 10$.

2. **Find the midpoints of each subinterval:**
- The first subinterval is $[-3, 7]$. The midpoint is $(-3+7)/2 = 2$.
- The second subinterval is $[7, 17]$. The midpoint is $(7+17)/2 = 12$.
- The third subinterval is $[17, 27]$. The midpoint is $(17+27)/2 = 22$.

**Solution:**
The x-coordinates at which the integrand is evaluated are $2, 12, 22$.

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