MHB Ahmed's question at Yahoo Answers regarding the midpoint rule

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The discussion centers on using the Midpoint Rule to approximate the integral of the function sqrt(x^4 + 19) from -2.5 to 1.5 with n=4. The Midpoint Rule formula is applied, leading to the calculation of midpoints for the specified intervals. The resulting approximation involves evaluating the function at these midpoints, yielding a final value of approximately 19.22. This method provides a numerical estimate for the integral, demonstrating the application of the Midpoint Rule in calculus. The calculations and steps are clearly outlined for clarity.
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Here is the original question:

Use the Midpoint Rule with n=4 to approximate?

1.5
Integral sqrt (x^4 +19) dx
-2.5

Here is a link to the question:

Use the Midpoint Rule with n=4 to approximate? - Yahoo! Answers

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

We are given to approximate:

$\displaystyle \int_{-2.5}^{1.5}\sqrt{x^4+19}\,dx$

using the midpoint rule with $n=4$

The Midpoint Rule is the approximation:

$\displaystyle \int_a^b f(x)\,dx\approx M_n$ where:

$\displaystyle M_n=\frac{b-a}{n}\cdot\sum_{k=0}^{n-1}\left[f\left(\frac{x_k+x_{k+1}}{2} \right) \right]$

where $\displaystyle x_k=a+k\frac{b-a}{n}$.

With $a=-2.5,\,b=1.5,\,n=4$ this becomes:

$\displaystyle M_n=\frac{1.5+2.5}{4}\cdot\sum_{k=0}^{3}\left[f\left(\frac{-2.5+k\frac{1.5+2.5}{4}-2.5+(k+1)\frac{1.5+2.5}{4}}{2} \right) \right]$

$\displaystyle M_n=\sum_{k=0}^{3}\left[f\left(k-2 \right) \right]$

Using the given integrand, this becomes:

$\displaystyle M_n=\sum_{k=0}^{3}\left[\sqrt{(k-2)^4+19} \right]$

$\displaystyle M_n=\sqrt{(0-2)^4+19}+\sqrt{(1-2)^4+19}+\sqrt{(2-2)^4+19}+\sqrt{(3-2)^4+19}$

$\displaystyle M_n=\sqrt{35}+\sqrt{20}+\sqrt{19}+\sqrt{20}=4\sqrt{5}+\sqrt{19}+\sqrt{35}\approx19.219250636639448$
 
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