We have used little vertically oriented rectangles to compute areas. There is no reason why little horizontal rectangles cannot be used. Here are the steps to find the area under the curve y = f(x) from x = 0 to x = 1 by using such horizontal rectangles.
a. Draw a picture with five horizontal rectangles, each of height 0.2, approximately filling the region to the right of the curve.
b. Calculate an upper and lower estimate of the length of each rectangle based on the length of the upper and lower boundaries.
c. Add these up to find upper and lower estimates of the area.
d. Think now of a very thin rectangle at height y. How long is the rectangle?
e. Write down a definite integral expression for the area.
f. Evaluate the integral and check that the answer is correct.
With f(x) = x2.