7.2 Calculus of parametric curves  (Page 5/6)

Therefore

One third of a second after the ball leaves the pitcher’s hand, the distance it travels is equal to

This value is just over three quarters of the way to home plate. The speed of the ball is

This speed translates to approximately 95 mph—a major-league fastball.

Surface area generated by a parametric curve

Recall the problem of finding the surface area of a volume of revolution. In Curve Length and Surface Area , we derived a formula for finding the surface area of a volume generated by a function $y=f\left(x\right)$ from $x=a$ to $x=b,$ revolved around the x -axis:

We now consider a volume of revolution generated by revolving a parametrically defined curve $x=x\left(t\right),y=y\left(t\right),a\le t\le b$ around the x -axis as shown in the following figure.

The analogous formula for a parametrically defined curve is

provided that $y\left(t\right)$ is not negative on $[a,b].$

Finding surface area

Find the surface area of a sphere of radius r centered at the origin.

We start with the curve defined by the equations

$x\left(t\right)=r\text{cos}t,y\left(t\right)=r\text{sin}t,0\le t\le \pi .$

This generates an upper semicircle of radius r centered at the origin as shown in the following graph.

When this curve is revolved around the x -axis, it generates a sphere of radius r . To calculate the surface area of the sphere, we use [link] :

$\begin{array}{cc}\hfill S& =2\pi {\displaystyle {\int }_a^{b}y\left(t\right)\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt}\hfill \\ & =2\pi {\displaystyle {\int }_0^{\pi }r\text{sin}t\sqrt{{\left(\text{−}r\text{sin}t\right)}^2+{\left(r\text{cos}t\right)}^2}dt}\hfill \\ & =2\pi {\displaystyle {\int }_0^{\pi }r\text{sin}t\sqrt{r^2{\text{sin}}^{2}t+r^2{\text{cos}}^{2}t}dt}\hfill \\ & =2\pi {\displaystyle {\int }_0^{\pi }r\text{sin}t\sqrt{r^2\left({\text{sin}}^{2}t+{\text{cos}}^{2}t\right)}dt}\hfill \\ & =2\pi {\displaystyle {\int }_0^{\pi }{r}^2\text{sin}tdt}\hfill \\ & =2\pi r^2({\text{−}\text{cos}t|}_0^{\pi })\hfill \\ & =2\pi r^2\left(\text{−}\text{cos}\pi +\text{cos}0\right)\hfill \\ & =4\pi r^2.\hfill \end{array}$

This is, in fact, the formula for the surface area of a sphere.

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Key concepts

• The derivative of the parametrically defined curve $x=x\left(t\right)$ and $y=y\left(t\right)$ can be calculated using the formula $\frac{dy}{dx}=\frac{y^{\prime }(t)}{x^{\prime }(t)}.$ Using the derivative, we can find the equation of a tangent line to a parametric curve.
• The area between a parametric curve and the x -axis can be determined by using the formula $A={\displaystyle {\int }_{t_1}^{t_2}y\left(t\right)x^{\prime }\left(t\right)dt}.$
• The arc length of a parametric curve can be calculated by using the formula $s={\displaystyle {\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}.$
• The surface area of a volume of revolution revolved around the x -axis is given by $S=2\pi {\displaystyle {\int }_a^{b}y\left(t\right)\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt}.$ If the curve is revolved around the y -axis, then the formula is $S=2\pi {\displaystyle {\int }_a^{b}x\left(t\right)\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt}.$

Key equations

• Derivative of parametric equations
  $\frac{dy}{dx}=\frac{dy\text{/}dt}{dx\text{/}dt}=\frac{y^{\prime }\left(t\right)}{x^{\prime }\left(t\right)}$
• Second-order derivative of parametric equations
  $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\left(d\text{/}dt\right)\left(dy\text{/}dx\right)}{dx\text{/}dt}$
• Area under a parametric curve
  $A={\displaystyle {\int }_a^{b}y\left(t\right)x^{\prime }\left(t\right)dt}$
• Arc length of a parametric curve
  $s={\displaystyle {\int }_{t_1}^{t_2}\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt}$
• Surface area generated by a parametric curve
  $S=2\pi {\displaystyle {\int }_a^{b}y\left(t\right)\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt}$

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.