How do you prove the area of a rectangle in Calculus?

Answer 1

First you would want to graph the rectangle. For example, The corners of the rectangle are (0,0), (0,2), (3,0) and (3,2). You would have a rectangle with the vertical sides being 2 units in length and the horizontal sides being 3 units.

The "easy" way to find the area of a rectangle is to multiple the length of the vertical sides by the horizontal sides. In this example, 2*3=6. The calculus way would be to setup an integral from a to b of f(x)dx. a and b are the end points for x values. i.e. a <= x <= b. In this example, a = 0 and b = 3. f(x) is the function y=2. The integral from 0 to 3 of 2dx = 6.