Math 106a Solutions
Quiz 2
9/24/10
1. Given the initial value problem dy
dx =โy2
xand y(1) = 1. Eulerโs method using step size โx= 0.25
produces the estimate y(2) โ0.5234. The graph shows a plot of the points compared to the curve
for the exact solution.
Recall: xn+1 =xn+ โxand yn+1 =yn+ โy
where โy= [slope at (xn,yn)] ยทโx.
x1 1.25 1.5 1.75 2
y0โ1โ0.45 โ0.2709 โ0.1855
y1 0.75 0.6375 0.5698 0.5234
2. Find the area of the region enclosed by the graphs of x=y2and x=โ2y2+ 3.
To find intersection points, solve y2=โ2y2+ 3
for y.
y2=โ2y2+ 3 โโ 3y2โ3=0 โโ
y=ยฑ1.
Therefore, the intersection points are (1,1) and
(1,โ1).
Set up the integral in terms of yto get:
Z1
โ1๎(โ2y2+ 3) โy2๎dy =Z1
โ1๎3โ3y2๎dy =h3yโy3i1
โ1
= 2 โ(โ2) = 4
Remark: to set up the integral in terms of x, the region needs to be separated into two pieces. The
total area is given by the following:
2Z1
0
โx dx + 2 Z3
1r3โx
2dx
Evaluate the second integral by substitution: u=3โx
2therefore du =โ1
2dx โโ โ2du =dx.
To change the limits of integration: if x= 1 then u= 1. Likewise, if x= 3, then u= 0.
2Z1
0
โx dx + 2 Z3
1r3โx
2dx = 2 Z1
0
x1/2dx โ4Z0
1
u1/2du = 2 ๎2
3x3/2๎1
0โ4๎2
3u3/2๎0
1
=4
3+8
3= 4
1