# Find the area of the region bounded

• ILoveBaseball
In summary, the conversation discusses finding the area of a region bounded by the equation r= 6-2sin(\theta). After determining the bounds for \theta, an integral is used to find the area. However, the obtained answer is incorrect and it is realized that the equation is actually a shifted and stretched ellipse, with the bounds always being 0 to 2pi.
ILoveBaseball
Find the area of the region bounded by: $$r= 6-2sin(\theta)$$

here's what i did:

$$6-2sin(\theta) = 0$$
$$sin(\theta) = 1/3$$

so the bounds are from arcsin(-1/3) to arcsin(1/3) right?

my integral:
$$\int_{-.339}^{.339} 1/2*(6-2sin(\theta))^2$$

i get a answer of 0.6851040673*10^11, and it's wrong. all my steps seems to be correct, i can't figure out the problem.

What is the answer that you should have got ?

marlon

$$r= 2sin(\theta)$$ is an ellipse so $$r= 6-2sin(\theta)$$ is just shifting and stretching it.

Therefore the bounds on $$\theta$$ are $$0 \leq \theta \leq 2 \pi$$

I agree.It's a shifted & stretched ellipse.Pay attention with the numbers...You can't get a big value for the area.It's ~100...

Daniel.

$38\pi$ to be exact.

asrodan said:
$$r= 2sin(\theta)$$ is an ellipse so $$r= 6-2sin(\theta)$$ is just shifting and stretching it.

Therefore the bounds on $$\theta$$ are $$0 \leq \theta \leq 2 \pi$$

can you explain it to me agian? i don't really understand it that well. are you saying if it's an ellipse, the bounds will always be from 0 ->2pi?

Yes,it's like for a circle,or for any closed curve enclosing the origin inside it...

Daniel.

ah, i get it now. thank you

## 1. How do I find the area of a region bounded by curves?

To find the area of a region bounded by curves, you can use the definite integral. First, identify the curves that form the boundaries of the region. Then, find the points of intersection between these curves. Finally, set up and evaluate the definite integral using these points as the limits of integration.

## 2. What is the formula for finding the area of a region bounded by curves?

The formula for finding the area of a region bounded by curves is ∫ab f(x) - g(x) dx, where f(x) and g(x) are the equations of the curves and a and b are the limits of integration.

## 3. Can I use any method other than integration to find the area of a bounded region?

Yes, if the region has a simple geometric shape like a rectangle, triangle, or circle, you can use the appropriate formula to find its area. However, if the region is bounded by curves, integration is the most accurate method to find its area.

## 4. How do I know if the region is bounded or not?

A region is considered bounded if it has a finite area. This means that the region is enclosed by a finite number of curves and does not extend to infinity. To determine if a region is bounded, you can graph the equations of the curves and check if they intersect and form a closed shape.

## 5. Can the area of a bounded region be negative?

No, the area of a region bounded by curves cannot be negative. The definite integral, which is used to find the area, only produces positive values. If you get a negative value when finding the area of a bounded region, it means that you have set up the integral incorrectly or the region is not bounded.

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