MATH 106A,B - CALCULUS II WINTER 2008
QUIZ 3
NAME:
Show ALL your work CAREFULLY.
(a) Consider the initial value problem
dy
dx = 1 + x2ywith y(0) = 2.
Estimate the value y(2) (when x= 2) of the solution using Euler’s method with two steps with initial point
(0,2). DO THIS BY HAND and show all your steps.
Since we use two steps from x= 0 to x= 2, the step size is ∆x= 1 with x0= 0 and x1= 1. Euler’s
method yields y1=y0+f(x0, y0)·∆xwhere f(x, y) = 1+x2y. It follows that y1= 2+[1+(0)2(2)](1) = 3.
Repeating the method again, we get y2=y1+f(x1, y1)·∆x= 3+[1+(1)2(3)](1) = 7. Thus, y(2) ≈7.
(b) Find the exact area of the region bounded by the curve x=y2+ 1, the line x= 1, and the line y= 1.
The area of the desired bounded region is given by
A=Z1
0
(y2+ 1) −1dy =y3
3¯
¯
¯
1
0=1
3.
If we integrate with respect to x, the area takes the form R2
11−√x−1dx.
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(c) Set up (but DO NOT evaluate) a definite integral representing the exact volume of the solid obtained
by rotating the region in (b) around the y-axis.
If we consider horizontal slices, the volume is represented by
Z1
0
[π(y2+ 1)2−π(1)2]dy =πZ1
0
y4+ 2y2dy.
or, integrating with respect to x, by
Z2
1
2πx(1 −√x−1) dx.
Date: January 28, 2008.
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