Estimate the area under the graph of f(x)=x2+2x from x=1 to x=7 using 3 approximating rectangles and left endpoints.

Answer 1

To divide the x-axis into 3 equal sections, subtract 7 and 1 and divide the result by 3:

7 - 1 = 6 and 6/3 = 2 so each rectangle will be 2 units wide. The left sides of the rectangles will be at x = 1, x = 3, and x = 5

The height of each rectangle is the function value for each of those x's:

f(1) = 1² + 2(1) = 3

f(3) = 3² + 2(3) = 15

f(5) = 5² + 2(5) = 35

The area of each rectangle is the width times the height so:

A₁ = (2)(3) = 6

A₂ = (2)(15) = 30

A₃ = (2)(35) = 70

Total area = A₁ + A₂ + A₃ = 6 + 30 + 70 = 106

You can see by this graph that this method will underestimate the total area.