c Dr Oksana Shatalov, Spring 2013
1
Spring 2013 Math 251
Week in Review 7
courtesy: Oksana Shatalov
(covering Sections 13.3(continued), 13.4-13.6 )
13.3: Double integrals over general regions (continued)
2
1. Sketch the region bounded by y 2 = 2x (or x = y2 ), the line x + y = 4 and the x-axis, in the
first quadrant. Find the area of the region using a double integral.
Z
2
Z
4
2. Describe the solid which volume is given by the integral
0
volume.
3. Find the volume of the solid bounded by the following planes
z = 1 + x + y, z = 0, x + y = 1, x = 0, y = 0.
y2
(x2 + y 2 )dx dy and find the
c Dr Oksana Shatalov, Spring 2013
2
13.4-13.5:Double integrals in polar coordinates
Key Points
• Let f be a continuous on the region D. Denote by D∗ the region representing D in the polar coordinates
(r, θ). Then
ZZ
ZZ
f (x, y) dA =
f (r cos θ, r sin θ) r drdθ.
D
D∗
4. Sketch the curve r2 = cos 2θ. Find the area inside the curve.
5. Use a double integral in polar coordinates to evaluate the area of the region inside the circle
r = 4 sin θ and outside the circle r = 2.
c Dr Oksana Shatalov, Spring 2013
3
6. Use polar coordinates to evaluate
√
Z
2
√
Z
√
− 2
4−x2
(x2 + y 2 ) dydx
|x|
7. Use polar coordinates to evaluate
Z
1
−1
Z √1−y2
√
−
ln(x2 + y 2 + 1) dxdy
1−y 2
RR
8. Change to polar coordinates in the double integral D f (x, y) dx dy where D is the region
bounded by the curve
(x2 + y 2 )2 = a2 (x2 − y 2 ), x ≥ 0
c Dr Oksana Shatalov, Spring 2013
4
9. Find the volume of the solid bounded by the surface z = 4 − x2 − y 2 and xy-plane (above
the plane).
10. Find the volume of the solid bounded by the surfaces
z=
p
1
64 − x2 − y 2 and z = (x2 + y 2 )
12
c Dr Oksana Shatalov, Spring 2013
5
13.6: Applications of double integral
Key Points
ZZ
• Total Mass m of the lamina with density ρ(x, y): m =
ρ(x, y) dA.
D
• Total charge Q: If an electric charge
Z Z is distributed over a region D and the charge density σ(x, y), then
the total charge Q is given by Q =
σ(x, y) dA.
D
ZZ
• Moment of the lamina with density ρ(x, y) that occupies the region D about the x-axis: Mx =
yρ(x, y) dA.
D
ZZ
• Moment of the lamina about the y-axis: Mx =
xρ(x, y) dA
D
RR
• Center of mass , (x̄, ȳ), of the lamina: x̄ =
D
xρ(x, y) dA
,
m
RR
ȳ =
D
yρ(x, y) dA
.
m
11. Find the mass of the plate bounded by x2 + y 2 = 4, x2 + y 2 = 9, if x ≤ 0 and y ≥ 0 if the
density is δ(x, y) = xy−4x
2 +y 2 .
12. Find the center of mass of the lamina that occupies the region bounded by the curves
y = x2 and y = x if the density at any point is proportional to the distance from the y-axis.