How are certain formulæ for areas of circles and ellipses related to calculus?

Answer 1

Area of an Ellispe of the form

#x^2/a^2+y^2/b^2=1#
can be found by #pi cdot a cdot b#, which can be viewed as a general form of the area of a circle since the equation of the circle
#x^2+y^2=r^2 Rightarrow x^2/r^2+y^2/r^2=1#,
which is an ellipse with #r=a=b#; therefore, the area of the circle is #picdot r cdot r=pir^2#.

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Answer 2

Unlike the area, the perimeter of an ellipse cannot easily be written down in closed form, like the formula for the circumference of a circle.

There is a whole area of advanced maths dealing with elliptic integrals. Not only does this give the perimeter of an ellipse as an infinite series, but leads on to solutions for the period of a simple pendulum with large amplitude, [such as here].(https://tutor.hix.ai)

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Answer 3

The formulas for the area of a circle ((A = \pi r^2)) and the area of an ellipse ((A = \pi ab)) are related to calculus through the concept of integration.

For the circle, the formula (A = \pi r^2) can be derived using calculus by considering the circle as a collection of infinitesimally thin concentric rings. Integrating the area of each ring from the center (radius 0) to the outer radius (r) gives the total area of the circle.

For the ellipse, the formula (A = \pi ab) can also be derived using calculus. One approach is to consider the ellipse as a stretched circle, with one radius (the major axis) longer than the other (the minor axis). By appropriately scaling the coordinates, the ellipse can be transformed into a circle, and its area can be calculated using the formula for the area of a circle.

In both cases, calculus provides the theoretical framework for understanding and deriving these area formulas, showing the connection between geometric shapes and the fundamental concepts of calculus, such as integration.

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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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