Homework week 10
Problems.
1. Consider region R given by the Semi-circle of radius r 1 in the upper-half of the xy-plane. Calculate
the integral of the function
2
2
f xy
ex y
over that given region R. Hint: change to polar coordinates. remember that:
Solution:
The integral changes like
e
x
r cos θ
y
r sin θ
dydx
rdrdθ
x2 y2
2
er rdrdθ
dydx
R
R
the limits of integration come easily from the sketch of the region R, Drawing a radial line we notice
y
y= 1−x2
r=1
1
θ=π
θ=0
−1
Figure 0.1: 0
that the r limits of integration go from r
to θ π so the integral changes to
e
x2
r
0 to r
y2
x
1
0
1, 0
θ
1, while the θ-limits of integration go from θ
dydx
R
π
π
0
1
2
er rdrdθ
0
π
1 r2
e
2
1
dθ
1
e 1 dθ
2
π
e 1
2
0
π
0
1
0
0
Notice to go from row 1 to row 2 we used the change of variable
so
1
r2
e rdr
r2
v
dv
2
rdr
ev dv
ev
er
2
0
e 1
1
0
2. Find the volume of the following figure: Solution:
z
1
y=x 2
1
1
y
x
Figure 0.2: fig1
Drawing a vertical line we find that the z-limits are: 01 dz. Drawing a line parallel to the y-axis we find
the y-limits: x12 dy and finally sliding the last line in the direction: ∞ to ∞ of the x-axis, we find that:
1
0 . Therefore the volume of this figure is given by
z
1
V
1
0
x2 0
1 1
0
x2
1 1
y
x2
0
1
0
1
1
dzdydx
1
0 dydx
dydx
1
x2 dx
1 x dx
2
13 x 1 2
1
3 3
0
1
3
x
0
3. Read problems 8.4 a) in these problems the volume was found by using double integrals methods. Using
triple integrals find the volumes of problems 8.4 a). Solution:
2
In this case drawing a vertical line will indicate that the z-limit goes from: 05 x y dz while the y and x
limits will remain the same as in 8.4 a). So the answer to this problem is
1
V
0
5 x2 y
x 1
0
0
2
dzdydx
the solution of the integral is as in problem 8.4 a) once you perform the first integral.
3