Math 1432
Section 15513
James West
[email protected]
620 PGH
Office Hours in 222 Garrison (CASA):
Monday 2:30 – 4:30 p.m.
Wednesday 2:30 – 4:30 p.m.
Area in Polar Coordinates
Section 9.5
The area of a polar region is based on the area of a sector of a
circle.
Area of a circle = πr2
Therefore the area of a sector of a circle is the part of the circle you
want times the area of the whole circle:
θ
1 2
2
•πr = r θ
Area sector =
2π
2
Find the area of the region between the origin and the polar graph
of r = ρ(θ) for θ between a and b.
The area of the region between the origin and the polar graph of
r = ρ(θ) for θ between a and b is given by
A =
b
∫a
2
2
1
1 b
 ρ (θ )  dθ = ∫  ρ (θ )  dθ
2
2 a
1. Find the area bounded by the graph of r = 2 + 2 sin θ.
2. Find the area inside one petal of the flower given by
r = 2 sin (3θ).
3. Find the area inside one petal of the flower given by
r = 4 cos (2θ).
For regions between two curves:
1
A =
2
b
∫a
  ρ (θ ) 2 −  ρ (θ ) 2  dθ
  1  
  2
1. Write the integral to find the area between r = 1 + cos θ,
r = cos θ, for θ = 0 to θ = π/2
2. Find the area inside r = 3 sin θ and outside r = 1 + sin θ.
3. Find the area between the loops of r = 1 + 2 cos θ.
2
4.
1
−3
−2
−1
1
−1
Give the area of the region that is in quadrant 4
and inside the outer loop of the polar graph
r = 1 – 2 cos (θ)
−2
1
−1
1
−1
POPPER 10
1. The polar plot of r = 2 + 2 cos θ is a
a. flower
b. line
c. cardioid
d. limaçon with loop
e. limaçon with dent (dimple)
2. The polar plot of r = 5 – 2 cos θ is a
a. flower
b. line
c. cardioid
d. limaçon with loop
e. limaçon with dent (dimple)
3. The polar plot of r = 7 – 12 cos θ is a
a. flower
b. line
c. cardioid
d. limaçon with loop
e. limaçon with dent (dimple)
4. The polar plot of r = 2 cos 5θ is a
a. flower with 5 petals
b. flower with 2 petals
c. flower with 10 petals
d. circle with radius 5
e. circle with diameter 2
5. The polar plot of r = 4 cos θ is a
a. circle centered at (0, 0)
b. flower with 4 petals
c. circle with radius 4, centered at (4, 0)
d. circle with radius 2, centered at (2, 0)
e. circle with radius 1, centered at (1, 0)
6. Give the integral that will determine the area inside one petal of
the flower given by r = sin (3θ).
a.
b.
π
3
1
2
∫ ( sin 3 θ ) dθ
0
π
3
∫ ( sin 3θ )
2
0
c.
d.
1
2
2
∫
π
3
0
π
6
dθ
( sin 3 θ ) dθ
∫ ( sin 3 θ )
2
2
0
e.
1
2
π
6
∫ ( sin 3θ )
0
2
dθ
dθ
Parametric Curves
Section 9.6
Three variables are used to represent a curve in the plane.
A parametric equation is usually an equation defined by some
parameter such as time.
Let x(t) and y(t) be functions where t is the parameter.
Then (x(t), y(t)) is the point that traces out the curve.
If t is restricted to lie on an interval [a, b] then x(t) and y(t) would
have an initial point (x(a), y(a)) and a terminal point (x(b), y(b)).
So a parametric curve has an orientation given by the
parameterizing variable.
Ex. 1:
Plot (cos(t), sin(t)) for 0 ≤ t ≤ 2π and express the curve by
an equation in x and y.
Ex. 2: Sketch the curve and eliminate the parameter.
x(θ) = 3 cos (θ)
y(θ) = 4 sin (θ)
0 ≤ θ ≤ 2π
Ex. 3: Give a parameterization of the PORTION of the line
y = –2x +5 between (1, 3) and (–2, 9)
Ex. 4: Parameterize a line segment from (a, b) to (c, d)
To parameterize a line SEGMENT from ( x 0 , y 0 ) to ( x 1 , y 1 ) :
x ( t ) = x 0 + t ( x1 − x 0 )
y ( t ) = y 0 + t ( y1 − y 0 )
0 ≤ t ≤1
For a LINE: −∞ ≤ t ≤ ∞
Ex. 5: Parameterize the line segment from (3, 6) to (–2, 5).
Ex. 6: Parameterize the line through (2, 1) and (8, 4)
Ex. 7: Express the curve by an equation in x and y; then sketch the
x ( t ) = 3t − 1 y ( t ) = 5 − 2t t ∈ ( −∞ , ∞ )
curve.
Ex. 8: Plot (t, t2) –1 ≤ t ≤ 1 and express the curve by an equation
in x and y.
Ex. 9: Plot (t, t3) –1 ≤ t ≤ 1 and express the curve by an equation
in x and y.
7. The parametric curve given by ( 2cos(t), 2sin(t) ) is a(n)
a. hyperbola
b. parabola
c. ellipse
d. circle
e. line
8. The parametric curve given by ( 3cos(t), 5sin(t) ) is a(n)
a. hyperbola
b. parabola
c. ellipse
d. circle
e. line
9. The parametric curve given by (t, 2t2 – 3t + 1) is a(n)
a. hyperbola
b. parabola
c. ellipse
d. circle
e. line