How do I show that the area of a circle is pi r^2 with integration?

Question

amistre64 · Answer

you intefrate the circumference :)

amistre64 · Answer

integrate even lol

amistre64 · Answer

2pi r from 0 to radius of circle

anonymous · Answer

integrate $$\sqrt{r ^{2}-x ^{2}}$$ from 0 to r . Multiply by 4

amistre64 · Answer

what that amounts to is adding up all the circumferences from 0 to the radius whih = area of the circle

amistre64 · Answer

$$\int_{0}^{r}2\pi.r dr$$

$$2\pi \int_{0}^{r}r.dr$$
$$2\pi \frac{r^2}{2}-2\pi \frac{0}{2}=\pi r^2$$

anonymous · Answer

Thank you soo much!!!!

amistre64 · Answer

youre welcome :)  i accidently discovered that when I tried finding the volume of a solid the wrong way lol

watchmath · Answer

Hi, Amistre
I think what you want to say is $$\int_0^{2\pi}r\,dr$$

anonymous · Answer

watchmath I integrated your formula, but I got 2 pi^2

watchmath · Answer

Ah sorry, it should be double integral
$\int_0^{2\pi}\int_0^r r\,drd\theta$
But I guess you haven't learn double integral

anonymous · Answer

not yet

watchmath · Answer

I think the more traditional one is the following
$$2\int_{-r}^r\sqrt{r^2-x^2}\,dx $$
But to compute this integral you need to use the trigonometric substitution.

anonymous · Answer

r is a constant right?

amistre64 · Answer

i typed it as i see it :)

$$2\pi \int_{0}^{r}r.dr$$
its the shell method for area instead of volume

amistre64 · Answer

area is the sum of all the circumferences of circles from 0 radius to full radius

amistre64 · Answer

top r might be better expreesed as x tho

watchmath · Answer

I see that know amistre :D

amistre64 · Answer

$$\frac{2\pi r^2}{2}|_{0}^{x}$$