Questions: Use the midpoint approximation to approximate the area under the curve of f(x) = x^2/2 + 2 on the interval [-4,2] using n=6 rectangles. Submit your answer using an exact value. For instance, if your answer is 10/3, then enter this fraction as your answer in the response box.

Use the midpoint approximation to approximate the area under the curve of f(x) = x^2/2 + 2 on the interval [-4,2] using n=6 rectangles.

Submit your answer using an exact value. For instance, if your answer is 10/3, then enter this fraction as your answer in the response box.
Transcript text: Question Use the midpoint approximation to approximate the area under the curve of $f(x)=\frac{x^{2}}{2}+2$ on the interval $[-4,2]$ using $n=6$ rectangles. Submit your answer using an exact value. For instance, if your answer is $\frac{10}{3}$, then enter this fraction as your answer in the response box.
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Solution

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Solution Steps

Step 1: Determine the width of each rectangle

The width of each rectangle, Δx=ban=2+46=1.\Delta x = \frac{b - a}{n} = \frac{2 + 4}{6} = 1.

Step 2: Midpoint Approximation

For the midpoint approximation, calculate the x-coordinate of the midpoint of each rectangle and the area of each rectangle. The general formula for the area of each rectangle is f(xmid,i)Δxf(x_{mid,i}) \cdot \Delta x, where xmid,i=a+(i0.5)Δxx_{mid,i} = a + (i-0.5)\Delta x. Sum up the areas of all rectangles to get the total approximate area under the curve.

Final Answer: The approximate area under the curve using the midpoint method with 6 rectangles is 23.75.

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