Section 4.2 Calculus 2 - Professor Clontz

4.2 Applications of Parametrizations

Let \(y\) be a function of \(x\), and suppose its curve is parametrized by the equations \(x(t),y(t)\). Then by the Chain Rule, \(\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\) at each point where all of these functions is differentiable.
Example Find the line tangent to the curve parametrized by \(x=\tan t,y=\sec t,-\frac{\pi}{2}<t<\frac{\pi}{2}\) at the point \((1,\sqrt 2)\).
Example Find the point on the parametric curve \(x=\ln t,y=t+\frac{1}{t},t>0\) which has a horizontal tangent line.

Suppose a curve \(C\) is defined parametrically by one-to-one functions \(x(t),y(t)\) on \(a\leq t\leq b\), where \(\frac{dx}{dt},\frac{dy}{dt}\) are continuous and never simultaneously zero. Then the length of \(C\) is defined to be \( L = \int_{t=a}^{t=b}ds = \int_{t=a}^{t=b}\sqrt{dx^2+dy^2} = \int_a^b\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\,dt \).
Example Use the arclength formula to find the length of the line segment joining \((-2,3)\) and \((2,0)\).

Example Find the perimeter of the curve parametrized by \(x=\sin^3 t,y=\cos^3 t,0\leq t\leq 2\pi\).

Suppose a smooth curve \(C\) is defined parametrically by one-to-one functions \(x(t),y(t)\) on \(a\leq t\leq b\), with \(y(t)\geq 0\). Then the area of the surface of revolution obtained by rotating \(C\) about the \(x\)-axis is given by \(2\pi\int_{t=a}^{t=b}y(t)\,ds\).

Example Find the area of the surface of revolution obtained by rotating the portion of the parabola \(y=x^2\) from \((0,0)\) to \((4,2)\) around the \(x\)-axis.

Find the line tangent to the curve parametrized by \(x=t^2,y=t^3\) at the point where \(t=-2\).
Show that the line tangent to the curve parametrized by \(x=3\sin t,y=3\cos t\) at the point \((\frac{3}{2},\frac{3\sqrt 3}{2})\) has the equation \(y=2\sqrt{3}-\frac{1}{\sqrt 3}x\). (Hint: \(\frac{1}{\sqrt 3}\frac{3}{2}=\frac{\sqrt 3}{2}\).)
Find the point on the parametric curve \(x=2t^2+1,y=t^4-4t\) which has a horizontal tangent line.
Use the arclength formula to find the length of the line segment joining \((-2,6)\) and \((3,-6)\).
Use the arclength formula to prove that the circumference of a circle of radius \(R\) is \(2\pi R\).
Show that the arclength of the curve parameterized by \(x=\cos 2t\), \(y=2t+\sin 2t\), \(0\leq t\leq \pi/2\) is \(4\). (Hint: \(1+\cos 2t=2\cos^2 t\).)
Find the area of the surface obtained by rotating the curve parameterized by \(x=\cos t,y=2+\sin t,0\leq t\leq \pi/2\) around the \(x\)-axis.
Use the parametric equations \(x=t,y=t,0\leq t\leq 1\) to show that the surface area of the cone of height \(1\) and radius \(1\) is \(\pi(\sqrt 2+1)\). (Hint: Don’t forget to add the area of the base of the cone.)
Show that the surface area of the cone of height \(H\) and radius \(R\) is \(\pi R(\sqrt{H^2+R^2}+R)\).
Find the point on the parametric curve \(x=e^{3t}+5,y=e^{2t}-2t+1\) which has a horizontal tangent line.
Find a definite integral that equals the arclength of the curve \(y=x^2-3x+4\) between \((1,2)\) and \((3,4)\).