Use a line integral to find the area of the region R. R: region inside the loop of the folium of...

Question:

Use a line integral to find the area of the region {eq}\mathrm{R} {/eq}.

{eq}\mathrm{R} {/eq}: region inside the loop of the folium of Descartes bounded by the graph of {eq}x = \displaystyle\frac{7t}{t^3 + 1} {/eq}, {eq}y = \displaystyle\frac{7t^2}{t^3 + 1} {/eq}.

Calculating the Area inside a Parametrically Defined Function

The formula for calculating the line integral of the function {eq}f(x, y) {/eq} over the smooth curve {eq}C {/eq} defined parametrically by the vector-valued function {eq}\mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} {/eq} over the interval {eq}a \leq t \leq b {/eq} is

{eq}\displaystyle \int_C f(x, y) \: dx = \int_a^b f(x(t), y(t)) \sqrt{[x'(t)]^2 + [y'(t)]^2} \: dt {/eq}

We can adjust this formula to find the area of a region {eq}\mathrm{R} {/eq} by setting {eq}f(x, y) = 1 {/eq}.

Answer and Explanation:

We will need to determine the limits of integration for the problem. In this problem we have {eq}x(t) = \displaystyle\frac{7t}{t^3 + 1} {/eq} and...

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Approximating Definite Integrals on a Graphing Calculator
Approximating Definite Integrals on a Graphing Calculator

from Saxon Calculus Homeschool: Online Textbook Help

Chapter 7 / Lesson 8
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