Estimate the area under the graph of f(x) = 3 + 4x^2 from x = −1 to x = 2 using six rectangles and midpoints.

Question

Estimate the area under the graph of f(x) = 3 + 4x^2 from x = −1 to x = 2 using six rectangles and midpoints.

Answer 1

The six rectangles you can use span these intervals on the x-axis: [-1, -0.5], [-0.5, 0], [0, 0.5], [0,5, 1], [1, 1.5], and [1.5, 2]. The midpoints of these intervals are -0.75, -0.25, 0.25, 0.75, 1.25, and 1.75 and each rectangle has a width of 0.5. So the estimated area using these rectangles are:

0.5 * f(-0.75) + 0.5 * f(-0.25) + 0.5 * f(0.25) + 0.5 * f(0.75) + 0.5 * f(1.25) + 0.5 * f(1.75)

Can you plug in the values of x into the function and evaluate this expression to get the estimate of the area?