Page 129 - Calculus Demystified

116
This is easily calculated to equal CHAPTER 4 The Integral
2 + 2 + 2 + 2 = 8.

2
3
You Try It: Calculate the (positive) area between y = x − 6x + 11x − 6 and
the x-axis.
EXAMPLE 4.10
Calculate the signed area between the graph of y = cos x + 1/2 and the
x-axis, −π/2 ≤ x ≤ π.
SOLUTION
This is easy, because the solution we seek is just the value of the integral:
π 1
Area = cos x + dx
−π/2 2

π
x
= sin x +
2
−π/2

π −π

= 0 + − −1 +
2 4
3π
= + 1.
4
Math Note: In the last example, we have counted positive area as positive and
negative area as negative. Our calculation shows that the aggregate area is positive.
We encourage the reader to draw a graph to make this result plausible.
2
You Try It: Calculate the actual positive area between the graph of y = x − 4,
−5 ≤ x ≤ 5 and the x-axis.
2
You Try It: Calculate the signed area between the graph of y = x − 4 and the
x-axis, −4 ≤ x ≤ 5.

4.4 The Area Between Two Curves

Frequently it is useful to find the area between two curves. See Fig. 4.17. Following
the model that we have set up earlier, we first note that the intersected region has
left endpoint at x = a and right endpoint at x = b. We partition the interval [a, b]
as shown in Fig. 4.18. Call the partition

P ={x 0 ,x 1 ,...,x k }.