## 17Calculus Parametric Equations - Surface Area

##### 17Calculus

On this page we discuss integration and surface area of parametric equations.

To calculate the surface area defined by revolving a parametric curve defined as $$x=X(t)$$ and $$y=Y(t)$$ from $$t=t_0$$ to $$t=t_1$$ about the x-axis, we use this integral.

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{ Y(t) \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~ dt}}$$

If you compare this integral to the equation for arc length (in the prevous section) you will see the common factor $$\sqrt{[X'(t)]^2 + [Y'(t)]^2}$$

Some textbooks write the surface integral differently, taking this into account. You may see it written as

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{Y(t) ~ ds}}$$
where $$ds = \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~dt$$

When rotating about the y-axis, the integral we use is

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{X(t) ~ ds}}$$

Notice we use a capital S to represent surface area. This is the standard symbol you will see in many textbooks. We reserve a lowercase s to represent arc length.

Practice

Find the surface area of revolution of the parametric curve $$x = t^4+4$$, $$y = 8t$$, $$0 \leq t \leq 2$$ rotated about the x-axis.

Problem Statement

Find the surface area of revolution of the parametric curve $$x = t^4+4$$, $$y = 8t$$, $$0 \leq t \leq 2$$ rotated about the x-axis.

Solution

### The Organic Chemistry Tutor - 3507 video solution

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Find the surface area of revolution of the parametric curve $$x = t^2$$, $$y = t^3$$, $$0 \leq t \leq 1$$ rotated about the y-axis.

Problem Statement

Find the surface area of revolution of the parametric curve $$x = t^2$$, $$y = t^3$$, $$0 \leq t \leq 1$$ rotated about the y-axis.

Solution

### The Organic Chemistry Tutor - 3508 video solution

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Find the surface area of revolution of the parametric curve $$x = 3t^2$$, $$y = 2t^3$$, $$0 \leq t \leq 5$$ rotated about the y-axis.

Problem Statement

Find the surface area of revolution of the parametric curve $$x = 3t^2$$, $$y = 2t^3$$, $$0 \leq t \leq 5$$ rotated about the y-axis.

$$\displaystyle{ S_y = \frac{24\pi}{5} \left[ 949\sqrt{26}+1 \right] }$$

Problem Statement

Find the surface area of revolution of the parametric curve $$x = 3t^2$$, $$y = 2t^3$$, $$0 \leq t \leq 5$$ rotated about the y-axis.

Solution

### Krista King Math - 594 video solution

video by Krista King Math

$$\displaystyle{ S_y = \frac{24\pi}{5} \left[ 949\sqrt{26}+1 \right] }$$

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Find the surface area of revolution of the parametric curve $$x = 1-t$$, $$y = 2\sqrt{t}$$, $$1 \leq t \leq 4$$ rotated about the x-axis.

Problem Statement

Find the surface area of revolution of the parametric curve $$x = 1-t$$, $$y = 2\sqrt{t}$$, $$1 \leq t \leq 4$$ rotated about the x-axis.

Solution

### Krista King Math - 59 video solution

video by Krista King Math

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