# Find Bounds for Area of Region: r^2 = 128cos(2\theta)

• ILoveBaseball
In summary, to find the area of the region bounded by r^2 = 128*cos(2\theta), you need to integrate 1/2*r^2 with respect to theta from 3pi/4 to 5pi/4, which gives a result of 128.
ILoveBaseball
Find the area of the region bounded by: $$r^2 = 128*cos(2\theta)$$

$$\int 1/2*(\sqrt{128*cos(2\theta)})^2$$

that's my integral, but i don't know what the bounds are. i tried typing it into my calculator, but i could only find one bound(.78539816) and I am not even sure if this is even correct. can someone help me find the bounds?

You know that cosine is periodic in 2pi, so theta can go from 0 to 2pi. Just by looking at it, you should be able to see that it will be a four leaf clover, and that each of the leaves will have the same area, so the total area will be 4 x the area of one. You should also be able to see that, since cosine is zero at pi/2 and 3pi/2, that one of the clover leafs will be from pi/4 to 3pi/4. And don't worry about r being negative, since the leaf created by taking r negative and theta going from pi/4 to 3pi/4 is just the leaf taking r positive going form 5pi/4 to 7pi/4, which you're already counting when taking the area of the one leaf and multiplying it by 4.

So your answer will be 4 x area of one leaf, where r is positive and theta ranges from pi/4 to 3pi/4.

ILoveBaseball said:
Find the area of the region bounded by: $$r^2 = 128*cos(2\theta)$$

$$\int 1/2*(\sqrt{128*cos(2\theta)})^2$$

that's my integral, but i don't know what the bounds are.

r^2 can not be negative, so

$$-\pi/4\le \theta \le \pi/4$$ and $$3\pi/4\le \theta \le 5\pi/4$$

ehild

Last edited:
How could you possibly get -64? I don't remember the formula for finding the area bounded by a curve given in polar co-ordinates, but if the formula you're using is correct, your integrand is strictly positive (it's a squared number (positive) times 1/2 (positive)).

Oh, I can't believe I didn't notice this. Something is wrong (maybe) since cosine will take on negative values on that region, but r² is positive, so I suppose it is implied that theta will only range over values where the equation makes sense. Cosine is non-negative from 3pi/2 to pi/2, so integrate from 3pi/4 to pi/4 (and not the other way around. Now, I'm guessing that if you're given r² = 128*cos(theta), maybe they want you to take the positive and negative values of r, so you'll get a 2-leafed clover. The area you need is then 2 x the area from 3pi/4 to pi/4 (of course, use 5pi/4, not pi/4), and that area is just 1/2r² d(theta), and r² is given to you, you don't have to write it as the square of a square root. You need to compute:

$$2\int _{\frac{3\pi }{4}} ^{\frac{5\pi }{4}} \frac{1}{2}r^2\, d\theta$$

$$= \int _{\frac{3\pi }{4}} ^{\frac{5\pi }{4}} 128\cos (2\theta)\, d\theta$$

$$= 64\sin (2\theta)|_ {\theta = \frac{3\pi }{4}} ^{\theta = \frac{5\pi }{4}}$$

$$= 64\left (\sin \left (\frac{pi}{2}\right ) - \sin \left (\frac{3\pi }{2}\right )\right )$$

$$= 128$$

## 1. What is the formula for finding the area of a region with the given equation?

The formula for finding the area of a region with the equation r^2 = 128cos(2θ) is A = (1/2)∫r^2dθ. This formula is derived from the polar coordinate area formula, which states that the area of a polar region is equal to half the integral of r^2 with respect to θ.

## 2. How do you find the bounds for the given equation?

To find the bounds for the given equation, you need to first solve for θ in terms of r. In this case, we can rewrite the equation as r = √(128cos(2θ)). Then, set r equal to 0 and solve for θ to find the lower bound. Next, set r equal to the maximum value of r (in this case, 8) and solve for θ to find the upper bound.

## 3. Can you explain the significance of the given equation in terms of the shape it represents?

The given equation, r^2 = 128cos(2θ), represents a cardioid shape. This can be visualized by plotting points with different values of θ and r, and connecting them to create a curve. This shape is commonly seen in nature, such as in the shape of a heart or the path of a planet orbiting around a star.

## 4. How can we use the given equation to find the area of a specific region?

To find the area of a specific region with the given equation, we need to determine the bounds for θ and then plug those values into the formula A = (1/2)∫r^2dθ. We can use a graphing calculator or software to help us visualize the region and determine the appropriate bounds. Once we have the bounds, we can evaluate the integral to find the area.

## 5. Are there any practical applications for finding the area of a region using this equation?

Yes, there are many practical applications for finding the area of a region using this equation. One example is in physics, where this equation can be used to calculate the moment of inertia for rotating objects with a cardioid shape. It can also be used in engineering and architecture to calculate the area of curved shapes, such as for designing bridges or buildings with unique shapes.

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