How do you find the area of circle using integrals in calculus?

Question

Isaiah Allen · 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#

Cameron Alexander · Answer

undefined To find the area of a circle using integrals in calculus, you can use the formula for the area of a circle, which is πr^2, where r is the radius of the circle. To derive this using integrals, consider dividing the circle into infinitely thin concentric rings of width dr. The area of each ring is approximately 2πr * dr. Integrating this expression from 0 to the radius of the circle gives the total area of the circle. Therefore, the integral becomes the integral from 0 to r of 2πr * dr, which simplifies to πr^2 when evaluated. Thus, the area of the circle can be found using the integral ∫(0 to r) 2πr dr = πr^2.