Given the parametric equations #x=acos theta# and #y=b sin theta#. What is the bounded area?

I got #-abpi# which is negative and clearly incorrect. What did I do wrong?

Answer 1

The sign is wrong as the parametric equations trace out in an anti-clockwise direction which will give a negative area,

The area in parametric coordinates is given by:

# A = int_alpha^beta \ y \ dx #

As we trace out in clockwise direction.

Now, we have parametric equations:

# { (x=acos theta), (y=b sin theta) :} => dx/(d theta) = -asin theta#

And these equations trace out in an anti-clockwise direction and so we must take account that; So the area of the ellipse is given by:

# A = -int_0^(2pi) \ (b sin theta)(-asin theta) \ d theta # # \ \ \ = ab \ int_0^(2pi) \ sin^2 theta \ d theta #

Using the identity:

# cos2A -= 1-2sin^2A => sin^2A -= 1/2(1-cos2A)#

Give us:

# A = ab \ int_0^(2pi) \ 1/2(1-cos2 theta) \ d theta # # \ \ \ = (ab)/2 \ int_0^(2pi) \ 1-cos2 theta \ d theta # # \ \ \ = (ab)/2 \ [theta-1/2sin2 theta ]_0^(2pi) # # \ \ \ = (ab)/2 \ {(2pi-1/2sin4pi) - (0-1/2sin0)} # # \ \ \ = (ab)/2 * 2pi # # \ \ \ = abpi #
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Answer 2

The bounded area enclosed by the curve described by the parametric equations (x = a\cos(\theta)) and (y = b\sin(\theta)) is a portion of the ellipse defined by those equations. The area can be calculated using the formula for the area of an ellipse, which is (A = \pi \times \text{{semi-major axis}} \times \text{{semi-minor axis}}), where the semi-major axis is (a) and the semi-minor axis is (b).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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