Calculus of Parametric Equations

Differentiation Of Parametric Curves

Full Derivative

1. Deparameterize
2. Derrive
3. Use the following equation to find x’(t) & y’(t) for the parametric equation

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$

Example

$$x(t)=2x+3,y(t)=3t-4$$

Solve first equation for t

$$t=\frac{x-3}{2}$$

Substitute into other equation

$$y=3(\frac{x-3}{2})-4$$ $$y=\frac{3}{2}x-\frac{9}{2}$$

Derive using the power rule, giving

$$\frac{dy}{dx}=\frac{3}{2}$$

Next use that equation to get x’(t) and y’(t)

$$\frac{dy}{dt}=3,\frac{dx}{dt}=2$$

Integrals Involving Parametric Equations

If the non-self-intersecting plane curve is defined by the parametric equations and x(t) is differentiable

$$x=x(t),y=y(t),a\le t\le b$$

Then the area under the curve is given by

$${\int }_a^{b}y(t)x^{\prime }(t)dt$$

1. Apply the above formula
2. integrate x(t) to get x’(t)
3. Solvar

Example

$$x(t)=t-sin(t),y(t)=1-cos(t),0\le t\le 2\pi$$ $$A={\int }_0^{2\pi }(1-cos(t))(\frac{d}{dt}(t-sin(t)))dt$$ $$A={\int }_0^{2\pi }(1-cos(t))(1-cos(t))dt$$ $$A={\int }_0^{2pi}1-2cos(t)+co{s}^2(t)dt$$

“Do some of that integration shit”