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.

Solution

Solution Steps

Step 1: Determine the width of each rectangle

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

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 $f(x_{mid,i}) \cdot \Delta x$, where $x_{mid,i} = a + (i-0.5)\Delta x$. Sum up the areas of all rectangles to get the total approximate area under the curve.