10 Parametric Equations and Polar Coordinates.
10.1 Curves Defined by Parametric Equations.
1. parametric equations (as equations of a third variable)
2. parametric equations with initial and terminal point
3. curve (set of points) versus parametric curve (curve with an orientation)
4. cartesian equation (equation of the form f(x1, x2,...xn) = 0. i.e. without the
extra parameter t. Example: x2+y2−1 = 0 or also x2+y2= 1)
5. cycloid (the curve traced out by a fixed point on the circumference of a rotating
circle along a straight line) with parametric equations x=r(θ−sin θ) and
y=r(1 −cos θ), with θ∈R
6. sketch parametric equations by combining xand yin order to eliminate the
third variable (tor θ). For the not so obvious ways to do so, we’ll use Section
10.2.
7. useful formulas in solving some trig integrals: sin2x=1−cos(2x)
2and cos2x=
1 + cos(2x)
2
8. length of an arc = rθ, where θis in radians
10.2 Calculus with Parametric Curves
1. tangents to parametric curves (slopes) dy
dx =
dy
dt
dx
dt
2. second derivative d2y
dx =
d(dy
dx )
dt
dx
dt
3. area A=Zb
a
ydx =Zβ
α
g(t)f0(t)dt, where x=f(t) and y=g(t) and α≤t≤β
when a≤x≤b
4. arc length L=Zb
ar1 + dy
dx2
dx =Zβ
αs1 + dy
dt
dx
dt 2dx
dt dt =Zβ
αrdx
dt 2
+dy
dt 2
dt
where x=f(t) and y=g(t) a nd α≤t≤βwhen a≤x≤b
5. surface area of the curve yrotating with the x-axis S=Zb
a
2πyds =Zβ
α
2πyrdx
dt 2
+dy
dt 2
dt
where x=f(t) and y=g(t) and α≤t≤βwhen a≤x≤b
