Estimate the area under the graph of f(x)= x^2 + 3 x from x=4 to x=8?

Question

using 4 approximating rectangles and left endpoints.

Answer 1

To estimate the area using rectangles, you need to first find the points where your rectangles start, then find the area of each rectangle. Since you're looking for the area from x = 4 to 8, using 4 rectangles at left endpoints, you need the value at x = 4, 5, 6 and 7. Each rectangle will thus have a width of 1.

Step 1: Find f(x) where x = 4, 5, 6, and 7.
f(4) = 4^2 + 3(4) = 28
f(5) = 5^2 + 3(5) = 40
f(6) = 6^2 + 3(6) = 54
f(7) = 7^2 + 3(7) = 70

Step 2: Using those points, find the area of the rectangles. The rectangles have a height defined by the answers above, and width of 1, so the area is equal to the height.

Step 3: Add the areas together for the total area.
A = 28 + 40 + 54 + 70
A = 192 (solution!)