Calculating Area with Green's Theorem If a simple closed...

Calculating Area with Green's Theorem If a simple closed curve C in the plane and the region $R$ it encloses satisfy the hypotheses of Green's Theorem, the area of $R$ is given by Green's Theorem Area Formula Area of $R=\frac{1}{2} \oint_{c} x d y-y d x$ The reason is that, by Equation (4) run backward, Area of $R=\iint_{k} d y d x=\iint_{k}\left(\frac{1}{2}+\frac{1}{2}\right) d y d x$ $$=\oint_{c} \frac{1}{2} x d y-\frac{1}{2} y d x$$ Use the Green's Theorem area formula given above to find the areas of the regions enclosed by the curves in Exercises $31-34$ The astroid $\mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, \quad \mathbf{0} \leq t \leq 2 \pi$

Calculating Area with Green's Theorem If a simple closed curve C in the plane and the region $R$ it encloses satisfy the hypotheses of Green's Theorem, the area of $R$ is given by
Green's Theorem Area Formula
Area of $R=\frac{1}{2} \oint_{c} x d y-y d x$
The reason is that, by Equation (4) run backward,
Area of $R=\iint_{k} d y d x=\iint_{k}\left(\frac{1}{2}+\frac{1}{2}\right) d y d x$
$$=\oint_{c} \frac{1}{2} x d y-\frac{1}{2} y d x$$
Use the Green's Theorem area formula given above to find the areas of the regions enclosed by the curves in Exercises $31-34$
The astroid $\mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, \quad \mathbf{0} \leq t \leq 2 \pi$

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Key Concepts

Green's Theorem

Green’s Theorem is a fundamental result in vector calculus that establishes the relationship between a line integral around a simple closed curve and a double integral over the region it encloses. This theorem enables the conversion of a complicated area calculation into a line integral along the boundary of the region, provided the vector field is continuously differentiable within the enclosed area.

Area Calculation Using Line Integrals

The area of a region enclosed by a simple closed curve can be computed via a line integral using the formula A = 1/2? (x dy - y dx). This approach leverages Green’s Theorem by expressing the double integral for area as a line integral, thereby simplifying the computation when the boundary of the region is described by a smooth, parameterized curve.

Parametric Representation of Curves

When curves are expressed in a parametric form, the coordinates of points on the curve are functions of a parameter, often simplifying the evaluation of integrals. This is particularly useful in applying the area formula derived from Green’s Theorem, as both the x and y coordinates and their differentials can be expressed in terms of the parameter, streamlining the process of calculating the area enclosed by the curve.

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