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

Question

Evelyn Agard · Accepted Answer

By using polar coordinates, the area of a circle centered at the origin with radius #R# can be expressed:
#A=int_0^{2pi}int_0^R rdrd theta=piR^2# Let us evaluate the integral,
#A=int_0^{2pi}int_0^R rdrd theta#
by evaluating the inner integral,
#=int_0^{2pi}[{r^2}/2]_0^R d theta=int_0^{2pi}R^2/2 d theta# 
by kicking the constant #R^2/2# out of the integral,
#R^2/2int_0^{2pi} d theta=R^2/2[theta]_0^{2pi}=R^2/2 cdot 2pi=piR^2#

Evelyn Agard · Answer

undefined 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.