# Section 4.2 Calculus 2

Applications of Parametrizations

## 4.2 Applications of Parametrizations

### 4.2.1 Parametric Formula for $$dy/dx$$

• 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.

### 4.2.2 Arclength

• 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 $$x=\sin^3 t,y=\cos^3 t,0\leq t\leq 2\pi$$.

### 4.2.3 Surface Areas from Revolution

• 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.

### Review Exercises

1. Find the line tangent to the curve parametrized by $$x=t^2,y=t^3$$ at the point where $$t=-2$$.
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: $$\frac{1}{\sqrt 3}\frac{3}{2}=\frac{\sqrt 3}{2}$$.)
3. Find the point on the parametric curve $$x=2t^2+1,y=t^4-4t$$ which has a horizontal tangent line.
4. Use the arclength formula to find the length of the line segment joining $$(-2,6)$$ and $$(3,-6)$$.
5. Use the arclength formula to prove that the circumference of a circle of radius $$R$$ is $$2\pi R$$.
6. 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: $$1+\cos 2t=2\cos^2 t$$.)
7. 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.
8. 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.)
9. Show that the surface area of the cone of height $$H$$ and radius $$R$$ is $$\pi R(\sqrt{H^2+R^2}+R)$$.
10. Find the point on the parametric curve $$x=e^{3t}+5,y=e^{2t}-2t+1$$ which has a horizontal tangent line.
11. Find a definite integral that equals the arclength of the curve $$y=x^2-3x+4$$ between $$(1,2)$$ and $$(3,4)$$.

Solutions

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 10.2, 6.3, 6.4