Math Problem Statement
Find the area of the region that is inside the region bounded by the graph of the first equation and outside the region bounded by the graph of the second equation.
r^2=4sen(2theta), r^2=2
Solution
To find the area of the region that is inside the region bounded by the graph of the first equation and outside the region bounded by the graph of the second equation , we need to proceed with the following steps:
Step 1: Interpret the equations
- The first equation describes a polar curve, specifically a lemniscate (a figure-eight shape).
- The second equation describes a circle in polar coordinates with radius .
Step 2: Determine the bounds of integration
To find the intersection points between the two curves, set the equations equal:
The solutions to within the interval are:
However, due to symmetry, we only need to consider one lobe of the lemniscate, i.e., from to .
Step 3: Compute the area
The area of the region between two polar curves and from to is given by:
For the region inside and outside , the area becomes:
Step 4: Evaluate the integral
To find the exact area, we need to compute the integral:
This can be split into two integrals:
First integral:
Second integral:
So:
[ A = 2 \left[-\frac{1}{2} \cos(2\theta)\right]{\frac{\pi}{12}}^{\frac{5\pi}{12}} - 2 \left[\theta\right]{\frac{\pi}{12}}^{\frac{5\pi}{12}} ]
Final Answer:
The area of the region inside the first curve and outside the second curve is .
Do you want more details or have any questions? Here are some related questions you might consider:
- How would the area change if the second curve was ?
- What if the first curve was a different polar curve, like ?
- Can you calculate the area of a single loop of the lemniscate ?
- What are some other applications of polar coordinates in calculating areas?
- How does the symmetry of polar curves simplify the area calculation?
Tip: When working with polar coordinates, symmetry can often simplify calculations by reducing the limits of integration.
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Math Problem Analysis
Mathematical Concepts
Polar Coordinates
Area Between Curves
Trigonometry
Formulas
Area between two polar curves formula
Theorems
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Suitable Grade Level
Undergraduate
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