Problem 0 In exercises find the area enclo... [FREE SOLUTION]

Chapter 9: Problem 0

In exercises find the area enclosed by the given curve. $$\left\\{\begin{array}{l} x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\ y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t \end{array}\right.$$

Short Answer

Expert verified

The area enclosed by the given curve is $蟺/2$.

Step by step solution

- Convert the equation into polar form

Given a parametric equation, the first step is to express the given function in the polar form, $r(t)$. From the trigonometric identity $cos(2t) = 1 - 2sin^2t$, we can write the x equation in terms of $sin^2(t)$. Similarly, for the y equation we can use the trigonometric identity $sin(2t) = 2sin(t)cos(t)$ to put it in terms of $sin(t)$ and $cos(t)$. Now the equations represent: $r(t) = 0.5 - 0.5sin^2(t)$ and $胃(t) = 0.5sin(t)cos(t) - 0.5sin^3(t)$.

- Derive the position function

Now that we have the position function, we can differentiate it to get the velocity function: $r'(t) = -0.5cos(t)sin(t)$ and $胃'(t) = 0.5cos^2(t) - 1.5sin^2(t)cos(t)$.

- Find the limit of parameter t

Find the limits of parameter t which form a closed curve. In this case the parameter t completes a full cycle from 0 to $2\pi$, expressed as $0 \leq t \leq 2\pi$.

- Integrate

Find the area enclosed by integrating the function over the defined limits. The area $A$ enclosed by the curve over a cycle from $蟿1$ to $蟿2$ is given by: $A = 0.5 * 鈭玙{蟿1}^{蟿2} [r(t) * r'(t) + 胃(t) * 胃'(t)] dt$. In the given case, the limits are $0$ and $2\pi$, so the integral is: $A = 0.5 * 鈭玙{0}^{2\pi} [0.5 - 0.5sin^2(t) * -0.5cos(t)sin(t) + 0.5sin(t)cos(t) - 0.5sin^3(t) * 0.5cos^2(t) - 1.5sin^2(t)cos(t)] dt$.

- Evaluate the integral

Evaluate the integral to get the area enclosed by the curve. The evaluation of the integral results in $A = \pi / 2$.

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91影视

In exercises find the area enclosed by the given curve. $$\left\\{\begin{array}{l} x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\ y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t \end{array}\right.$$

Short Answer

Step by step solution

- Convert the equation into polar form

- Derive the position function

- Find the limit of parameter t

- Integrate

- Evaluate the integral

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