Name: Solutions
Math 53: Quiz 2
September 9, 2014
1. (1 point) Find the Cartesian equation of the curve and sketch it: x= cos(θ), y =
1 + sin(θ),0≤θ < 2π.
We have cos(θ) = xand sin(θ) = y−1. Thus,
cos2(θ) + sin2(θ) = 1 ⇒x2+ (y−1)2= 1.
This represents a circle of radius 1 centered at (0,1).
−3 −2 −1 0 1 2 3
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 1
2. (1 point) Find the length of the curve defined by: x=t2
2, y =(2t+1)3/2
3,0≤t≤4.
Note that dx
dt =tand dy
dt = (2t+ 1)1/2. We then have from the arc length formula
L=∫t=4
t=0 √(t)2+ (√2t+ 1) dt
=∫t=4
t=0
√t2+ 2t+ 1 dt
=∫t=4
t=0 √(t+ 1)2dt
=∫t=4
t=0
t+ 1 dt (∵t≥0⇒ |t+ 1|=t+ 1)
=[t2
2+t]4
0
=[16
2+ 4]−0 = 12.
1
