22 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
Example 1.27 Find the area between y D x
2
and y D x C 2.
Solution:
A
3-3
-1
5
y D x
2
y D x C2
Solving
x
2
D x C2;
or equivalently
x
2
x 2 D 0;
we see that .x 2/.x C 1/ D 0. So x D 1; 2, making the intersection points .1; 1/ and .2; 4/.
is also gives us the limits of integration and so
1.2. THE FUNDAMENTAL THEOREM 23
A D
Z
2
1
x C2 x
2
dt
D
Z
2
1
2 Cx x
2
dt
D 2x C
1
2
x
2
1
3
x
3
ˇ
ˇ
ˇ
ˇ
2
1
D 4 C
4
2
8
3
C 2
1
2
1
3
D
24 C12 16 C12 3 2
6
D 27=6
D 9=2 units
2
˙
Let’s do an example that moves beyond polynomial functions.
Example 1.28 Find the area under the sine function from x D 0 to x D .
Solution:
A
4
-1
-1
1
y D sin.x/
x D
x D 0
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