For each problem, approximate the area under the curve over the given interval using 5 midpoint rectangles. 8) y = -x + 5; [-3, 2]

Question

Riemann Sum

Answer 1

If it helps, here is a sketch of what Juliya K. explained.

Answer 2

Since there will be 5 rectangles over the interval [-3, 2], each rectangle will have a width of:

(2 – -3) ⁄ 5 = 5/5 = 1

This uses the formula (b – a)/n to find the width of n rectangles over any interval [a , b]. Thus:

1. The first rectangle with be from x = -3 to x = -2, so the midpoint is at x = -2.5
2. The second rectangle will be from x = -2 to x = -1, so the midpoint is at x = -1.5
3. The third rectangle will be from x = -1 to x = 0, so the midpoint is at x = -0.5
4. The fourth rectangle will be from x = 0 to x = 1, so the midpoint is at x = 0.5
5. The fifth rectangle will be from x = 1 to x = 2, so the midpoint is at x = 1.5

Now that we know where the midpoint is for each of the five rectangles, we can use that to find the height for every rectangle. Using the equation y = -x + 5, we get:

1. The first rectangle has a height of y = -(-2.5) + 5 = 7.5
2. The second rectangle has a height of y = y = -(-1.5) + 5 = 6.5
3. The third rectangle has a height of y = y = -(-0.5) + 5 = 5.5
4. The fourth rectangle has a height of y = y = -(0.5) + 5 = 4.5
5. The fifth rectangle has a height of y = y = -(1.5) + 5 = 3.5

Since the area of a rectangle is given by A = width * height, we can put everything together and then add all the areas. However, since every rectangle has a width of 1, we know that all the areas will equal the heights we found in the last step. Therefore the approximated area under the curve is simply

7.5 + 6.5 + 5.5 + 4.5 + 3.5 = 27.5