A curve is given by the following parametric equations. x(T)=2cos(t), y(t)=7cos(t)sin(t), 0 is less than or equal to t is less than or equal to 2pi. Use symmetry to Find the area bounded by the curve over the given interval

To find the area enclosed by the curve given by the parametric equations:

\[
x(t) = 2 \cos(t), \quad y(t) = 7 \cos(t) \sin(t)
\]

over the interval \(0 \leq t \leq 2\pi\), we can use the formula for the area enclosed by a parametric curve:

\[
A = \int_{a}^{b} y(t) \frac{dx}{dt} \, dt
\]

Here, \(x(t) = 2 \cos(t)\) and \(y(t) = 7 \cos(t) \sin(t)\), and the interval is from \(0\) to \(2\pi\).

### Step 1: Calculate \( \frac{dx}{dt} \)

\[
\frac{dx}{dt} = \frac{d}{dt} (2 \cos(t)) = -2 \sin(t)
\]

### Step 2: Set up the integral

Substitute \(y(t)\) and \( \frac{dx}{dt}\) into the formula for the area:

\[
A = \int_{0}^{2\pi} 7 \cos(t) \sin(t) (-2 \sin(t)) \, dt
\]

This simplifies to:

\[
A = -14 \int_{0}^{2\pi} \cos(t) \sin^2(t) \, dt
\]

### Step 3: Use symmetry to simplify the calculation

Notice that \( \cos(t) \sin^2(t) \) is an odd function over the interval \(0 \leq t \leq 2\pi\). Since the function is symmetric about \( \pi\), the integral over the full period will be zero:

\[
\int_{0}^{2\pi} \cos(t) \sin^2(t) \, dt = 0
\]

Thus, the area enclosed by the curve is:

\[
A = 0
\]

The curve’s symmetry results in the net area calculation canceling out to zero over one complete period from \(0\) to \(2\pi\).

Would you like further details, or do you have any questions? Here are some related questions to consider:

1. How do we determine if a parametric curve encloses any area?
2. What role does symmetry play in evaluating integrals of trigonometric functions?
3. Can the area calculation for a parametric curve be done using polar coordinates?
4. How does the parametrization of a curve affect its geometric properties?
5. What are the conditions for using Green’s Theorem to find the area of a parametric curve?

**Tip:** When working with trigonometric integrals, always consider symmetry to simplify the calculations.

A curve is given by the following parametric equations. x(t)=2cos(t), y(t)=7cos(t)sin(t), 0 ≤ t ≤ 2π. Use symmetry to find the area bounded by the curve over the given interval.

Explore how to find the area bounded by the parametric curve defined by x(t)=2cos(t) and y(t)=7cos(t)sin(t) over the interval 0 ≤ t ≤ 2π. Learn about the use of symmetry and the integral of trigonometric functions. This solution is ideal for students in Grades 11-12 and early calculus courses.

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