Page 120 - Calculus Demystified

The Integral
CHAPTER 4
For our first example, we calculate the area under a parabola. 107
EXAMPLE 4.4
2
Calculate the area under the curve y = x , above the x-axis, and between
x = 0 and x = 2.
SOLUTION
2
Refer to Fig. 4.8 as we reason along. Let f(x) = x .

Fig. 4.8

Consider the partition P of the interval [0, 2] consisting of k + 1 points
x 0 ,x 1 ,...,x k . The corresponding Riemann sum is
k

R(f, P) = f(x j ) · x.
j=1
Of course
2 − 0 2
x = =
k k
and
2
x j = j · .
k
In addition,
2 2
2 4j
f(x j ) = j · = .
k k 2