Applications of the Initial Value Problem y' = J(x, y); y(c) = d 1253 y2x
-3 -3 -2 -1^0 2 3
-1-2-3 -3Figure 3. 5 Rose Curve r = 3 sin 28 Figure 3.6 Lemniscate r^2 = 9 sin 28
EXERCISES 3.1In the following exercises approximate 7f by 3.141593 when nec-
essary.
In exercises 1-6 use SOLVEIVP or your computer software to
evaluate numerically the given definite integrals.
l. J~ 1 ;/x^3 +1 dx 2. J1 -1 ;/1 + e-^2 x dx
- J3 1 e -x2 d x 4. fo2 xx dx
5. J3 -1 ex x dx 6. f 0 1r ;/1 + cosxdx
In exercises 7-12 use SOLVEIVP or your computer software to
calculate the area under the given curve y = f(x) over the given
interval [a , b].
7. j(x ) = 1/ lnx on [2, 3]
- j(x)=xtan x on[0, 7r/4]
11. j(x ) = 1/;/1 - x^3 on [-1, OJ
8. J(x)=exlnx on[.5,2]
10. J(x) = ln(tanx) on [O, 7r/3]
- f(x) = ;/1 + sinx on [O, 7r]
In exercises 13-16 numerically calculate the area of the region
bounded by the given sets of curves.
13. y = ;/2 + x^2 , y = x^2 , x = 0
- y = Vf+"X, y = (x + 1)^2