Using Green's Theorem, calculate the area of the indicated region.
-The astroid $\mathbf { r } ( \mathrm { t } ) = \left( 5 \cos ^ { 3 } \mathrm { t } \right) \mathbf { i } + \left( 5 \sin ^ { 3 } \mathrm { t } \right) \mathbf { j } , 0 \leq \mathrm { t } \leq 2 \pi$
A) $\frac { 75 } { 8 } \pi$
B) $\frac { 75 } { 4 } \pi$
C) $2 \pi$
D) $\frac { 75 } { 16 } \pi$

Question

Quizplus · Accepted Answer

The area $A$ enclosed by the curve given by the parametric equations $x = 5\cos^3(t)$ and $y = 5\sin^3(t)$ can be calculated using Green's Theorem by integrating $dA = dx \, dy$. The formula for the area using Green's Theorem in this context simplifies to $A = \int_C x \, dy$, where $C$ is the curve defined by the parametric equations. Substituting $x = 5\cos^3(t)$ and $dy = 15\cos^2(t)\sin(t) \, dt$, and integrating from $0$ to $2\pi$, we obtain $A = \frac{75}{8}\pi$.

Using Green's Theorem, Calculate the Area of the Indicated Region