replace the values of with the appropriate widths of each rectangle. The width of the first
rectangle is 2 − 0 = 2; the second width is 5 − 2 = 3; the third is 11 − 5 = 6; the fourth is 19 − 11 = 8; the
fifth is 22 − 19 = 3; and the sixth is 23 − 22 = 1. We find the height of each rectangle by evaluating f(x) at
the appropriate value of x, the left endpoint of each interval on the x-axis. Here, y 0 = 4, y 1 = 6, y 2 = 16, y 3
= 18, y 4 = 22, and y 5 = 29. Therefore, we can approximate the integral with f(x) dx = (2)(4) + (3)(6)
- (6)(16) + (8)(18) + (3)(22) + (1)(29) = 361.
That’s all there is to approximating the area under a curve using rectangles. Now let’s learn how to find
the area exactly. In order to evaluate this, you’ll need to know...
The Fundamental Theorem of Calculus
Before, we said that if you create an infinite number of infinitely thin rectangles, you’ll get the area under
the curve, which is an integral. For the example above, the integral is
x^3 dxThere is a rule for evaluating an integral like this. The rule is called the Fundamental Theorem of
Calculus, and it says
f(x) dx = F(b) − F(a); where F(x) is the antiderivative of f(x).Using this rule, you can find x^3 dx by integrating it, and we get . Now all you do is plug in 3 and 2
and take the difference. We use the following notation to symbolize this
Thus, we have