# How do you find the area of a circle using integration?

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To find the area of a circle using integration, you can consider a circle of radius r. The equation of the circle is x^2 + y^2 = r^2. Solving for y, we get y = sqrt(r^2 - x^2).

The area of the circle can be found by integrating the function y = sqrt(r^2 - x^2) from -r to r with respect to x. This integral represents the area under the curve of the upper half of the circle. However, since the circle is symmetric about the y-axis, we can find the total area by doubling this integral.

Therefore, the area A of the circle is given by:

A = 2 * ∫[from -r to r] sqrt(r^2 - x^2) dx.

This integral can be solved using a trigonometric substitution or by recognizing it as the formula for the area of a semicircle and doubling it to get the area of the full circle.

The integral evaluates to:

A = 2 * (πr^2) / 2 = πr^2.

So, the area of a circle with radius r is πr^2.

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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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