MHB Kendra N's question at Yahoo Answers regarding the Midpoint Rule

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The discussion focuses on using the Midpoint Rule to approximate the integral of 2 cos^5(x) from 0 to π/2 with n set to 4. The formula for the Midpoint Rule is presented, leading to the calculation of M_4, which involves evaluating the function at specific midpoints. The final approximation calculated is M_4 ≈ 1.0667, rounded to four decimal places. Additionally, the true value of the integral is provided for comparison, which is approximately 1.0667. This demonstrates the effectiveness of the Midpoint Rule in estimating definite integrals.
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Here is the question:

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal?

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places

∫ 2 cos^5 (x) dx from x = 0 to x = π/2, n = 4

Here is a link to the question:

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Kendra N,

The Midpoint Rule is the approximation $\displaystyle \int_a^b f(x)\,dx\approx M_n$ where:

$\displaystyle M_n=\frac{b-a}{n}\sum_{k=1}^n\left[f\left(\frac{x_{k-1}+x_k}{2} \right) \right]$

We are asked to use this rule to approximate:

$\displaystyle 2\int_0^{\frac{\pi}{2}}\cos^5(x)\,dx$

Identifying:

$\displaystyle a=0,\,b=\frac{\pi}{2},\,n=4,\,f(x)=\cos^5(x),\,x_k=k\cdot\frac{\pi}{8}$, we have:

$\displaystyle M_4=2\frac{\frac{\pi}{2}-0}{4}\sum_{k=1}^4\left[\cos^5\left(\frac{(k-1)\cdot\frac{\pi}{8}+k\cdot\frac{\pi}{8}}{2} \right) \right]$

This simplifies to:

$\displaystyle M_4=\frac{\pi}{4}\sum_{k=1}^n\left[\cos^5\left(\frac{\pi}{16}(2k-1) \right) \right]$

Using a calculator/computer and rounding to 4 decimal places, we find:

$\displaystyle M_4\approx1.0667$

For comparison, the true value of the integral is:

$\displaystyle 2\int_0^{\frac{\pi}{2}}\cos^5(x)\,dx=\frac{16}{15}=1.0\bar{6}$.
 
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