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Expert-verified Found in: Page 535 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # (a) How do you find the slope of a tangent to a parametric curve? (b) How do you find the area under a parametric curve?

1. For a parametric curve $${\rm{x = f(t), y = g(t),}}$$ the slope of the tangent is $$\frac{{{\rm{g'(t)}}}}{{{\rm{f'(t)}}}}$$.
2. For a parametric curve $${\rm{x = f(t), y = g(t),}}$$the area under the curve is $${\rm{A = }}\int_{\rm{\alpha }}^{\rm{\beta }} {{\rm{g(t)f'(t)dt}}}$$.
See the step by step solution

## Step 1: Concept Introduction

Parametric curves allow us to draw relationships between two or more numbers while also representing the directions or orientations of each quantity.

## Step 2: Slope of tangent to Parametric Curve

(a)

The parametric curve is represented as –

$${\rm{x = f(t), y = g(t)}}$$

Differentiate the functions with respect to $${\rm{t}}$$–

$$\frac{{{\rm{dx}}}}{{{\rm{dt}}}}{\rm{ = f'(t), }}\frac{{{\rm{dy}}}}{{{\rm{dt}}}}{\rm{ = g'(t)}}$$

Now use the chain rule to obtain the slope –

Slope of tangent$${\rm{ = }}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}$$

\begin{aligned}{c}{\rm{ = }}\frac{{{\rm{dy}}}}{{{\rm{dt}}}} \cdot \frac{{{\rm{dx}}}}{{{\rm{dt}}}}\\{\rm{ = }}\frac{{{{{\rm{dy}}} \mathord{\left/ {\vphantom {{{\rm{dy}}} {{\rm{dt}}}}} \right. \kern-\nulldelimiterspace} {{\rm{dt}}}}}}{{{{{\rm{dx}}} \mathord{\left/ {\vphantom {{{\rm{dx}}} {{\rm{dt}}}}} \right. \kern-\nulldelimiterspace} {{\rm{dt}}}}}}{\rm{ = }}\frac{{{\rm{g'(t)}}}}{{{\rm{f'(t)}}}}\end{aligned}

Therefore, the result is obtained as $$\frac{{{\rm{g'(t)}}}}{{{\rm{f'(t)}}}}$$.

## Step 3: Area under a Parametric Curve

(b)

The area under the curve $${\rm{y = F(x)}}$$ from $${\rm{a}}$$ to $${\rm{b}}$$ is given by $${\rm{A = }}\int_{\rm{a}}^{\rm{b}} {{\rm{F(x)dx}}}$$ where $${\rm{F(x)}} \ge {\rm{0}}$$.

If curve is given by parametric equations –

$${\rm{x = f(t), y = g(t), \alpha }} \le {\rm{t}} \le {\rm{\beta }}$$

Then using previous formula for the area under parametric curve, it is obtained that –

$${\rm{A = }}\int_{\rm{a}}^{\rm{b}} {{\rm{ydx}}} {\rm{ = }}\int_{\rm{\alpha }}^{\rm{\beta }} {{\rm{g(t)f'(t)dt}}}$$

Therefore, the result is obtained as $${\rm{A = }}\int_{\rm{\alpha }}^{\rm{\beta }} {{\rm{g(t)f'(t)dt}}}$$. ### Want to see more solutions like these? 