5.3
Area and Definite Integrals
3 2 1 0 -1 2 0 -2 -4 x y x yTheorem 61 (The definite integral = Area of region) If is continuous and
nonnegative in the [ ] then the area of the region bounded is Area =
Z
()
In particular R () ≥ 0
Example 127 Find the area of the region between the graph of
() = 2− 1
2
2
and the −axis from = −1 to = 2
2.5 1.25 0 -1.25 -2.5 3 2 1 0 -1 x y x y 68
Theorem 62 If is continuous in the [ ] then the area of the region bounded is
Area =
Z
| ()|
Example 128 Find the area of the region between the graph of
() =−2+ 4− 3
and the −axis from = 0 to = 2
3 2.5 2 1.5 1 0.5 0 1 0 -1 -2 -3 x y x y
Theorem 63 If and are integrable in [ ] and () ≤ () for all
∈ [ ] then Z () ≤ Z ()
Theorem 64 If and are integrable in [ ] and 0 ≤ () ≤ () for all
∈ [ ] then the area of the region bounded by the graph of () and ()
on [ ] is
Z
[ ()− ()]
Example 129 Find the area of the region bounded by the graph of = 22
− 3 + 2 the -axis
Example 130 Find the area of the region bounded by the graph of = 2−2 and = 1.25 0 -1.25 -2.5 2 0 -2 -4 -6 x y x y
Example 131 Find the area of the region bounded by the graph of = 2
and = 2 − 2 1.5 1 0.5 0 -0.5 -1 1.25 0 -1.25 -2.5 x y x y 70
