Estimate the area under the graph of f(x)= x^2 + 2 x from x=4 to x=10 using 3 approximating rectangles and left endpoints.

not clear what you mean by "left endpoints"

I would mark off x= 4,6,8, and 10
f(4), f(6), .. etc would give you the heights
Do you want the rectangles to overlap or underlap?
rectangle 1:
larger =2 x f(6) = 2(48) = 96
smaller = 2(f(4)) = 48
difference = 48, which cut in half is 24
so first rectangle estimation = 48+24 = 72
do the same for the other two

I got 128 and 200 for a total
estimation of 400

BTW, using integration gave me an exact area of 396

To estimate the area under the graph of f(x) = x^2 + 2x from x = 4 to x = 10 using 3 approximating rectangles and left endpoints, follow these steps:

1. Determine the width of each rectangle: Calculate the width (delta x) by subtracting the starting x-value from the ending x-value and dividing by the number of rectangles. In this case, (10 - 4) / 3 = 2.

2. Identify the left endpoints of each rectangle: Start with the first x-value, which is 4, and for each rectangle after that, add the width (delta x) to get the next left endpoint. In this case, the left endpoints are 4, 6, and 8.

3. Evaluate the function at the left endpoints: Substitute each left endpoint into the function f(x) = x^2 + 2x to get the corresponding y-values. Calculate f(4), f(6), and f(8).

f(4) = 4^2 + 2 * 4 = 16 + 8 = 24
f(6) = 6^2 + 2 * 6 = 36 + 12 = 48
f(8) = 8^2 + 2 * 8 = 64 + 16 = 80

4. Calculate the area of each rectangle: Multiply the width (delta x) by the height (y-value) for each rectangle to get its area.

Rectangle 1: Area = (2) * (24) = 48
Rectangle 2: Area = (2) * (48) = 96
Rectangle 3: Area = (2) * (80) = 160

5. Sum up the areas of all the rectangles: Add the areas of all three rectangles together.

Total area ≈ 48 + 96 + 160 = 304

Therefore, the estimated area under the graph of f(x) = x^2 + 2x from x = 4 to x = 10 using 3 approximating rectangles and left endpoints is approximately 304 square units.