An ellipse having x = a cos θ and y = b sin θ. Calculate area of the ellipse.

ANSWER : To calculate the area of the ellipse defined by the parametric equations x = a cos θ and y = b sin θ, we can use the formula for the area of an ellipse, which is :

Area = πab

where 'a' is the semi-major axis and 'b' is the semi-minor axis of the ellipse.

In this case, the parametric equations x = a cos θ and y = b sin θ represent the ellipse in standard form centered at the origin (0, 0) with the semi-major axis 'a' along the x-axis and the semi-minor axis 'b' along the y-axis.

To find 'a' and 'b,' we can relate the parametric equations to the standard form of an ellipse :

x = a cos θ → a = x / cos θ

y = b sin θ → b = y / sin θ

Now, to find the limits of integration for θ, we need to determine where one complete cycle of the ellipse lies. Since the parametric equations x = a cos θ and y = b sin θ represent one complete cycle of the ellipse when θ ranges from 0 to 2π, we will integrate the area formula over this range of θ.

Area = \(\int_0^{2\pi}\)(πab) dθ

Area = πab \(\int_0^{2\pi}\) dθ

Area = πab \(\left[\theta\right]_0^{2\pi}\)

Area = πab (2π - 0)

Area = 2π²ab

Therefore, the area of the ellipse defined by the parametric equations x = a cos θ and y = b sin θ is 2π²ab.

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