Area of Circle

x2+y2=r2y2=r2x2y=±r2x2\begin{aligned} x^2+y^2&=r^2\\ y^2&=r^2-x^2\\ y&=\pm\sqrt{r^2-x^2} \end{aligned}

A=rrr2x2dx+rrr2x2dx=rrr2x2dx+rrr2x2dx=rrr2x2dx+rrr2x2dx=2rrr2x2dx, let x=rsinθdx=rcosθdθ=2π2π2r2(rsinθ)2rcosθdθ=2π2π2r2r2sin2θrcosθdθ=2π2π2r2(1sin2θ)rcosθdθ=2π2π2r2cos2θrcosθdθ=2π2π2rcosθrcosθdθ=2π2π2r2cos2θdθ=2r2π2π2(12+12cos2θ)dθ=2r2[12θ+14sin2θ]π2π2=2r2(12(π2)+14sin(2(π2))(12(π2)+14sin(2(π2))))=2r2(π4+14sinπ+π4+14sinπ)=2r2(π2)A=πr2\begin{aligned} A&=\int_{-r}^{r}\sqrt{r^2-x^2}dx+\left|\int_{-r}^{r}-\sqrt{r^2-x^2}dx\right|\\ &=\int_{-r}^{r}\sqrt{r^2-x^2}dx+\left|-\int_{-r}^{r}\sqrt{r^2-x^2}dx\right|\\ &=\int_{-r}^{r}\sqrt{r^2-x^2}dx+\int_{-r}^{r}\sqrt{r^2-x^2}dx\\ &=2\int_{-r}^{r}\sqrt{r^2-x^2}dx\text{, let }x=r\sin\theta\Rightarrow dx=r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{r^2-(r\sin\theta)^2}r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{r^2-r^2\sin^2\theta}r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{r^2\left(1-\sin^2\theta\right)}r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{r^2}\sqrt{\cos^2\theta}r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}r\cos\theta r\cos\theta d\theta\\ &=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}r^2\cos^2\theta d\theta\\ &=2r^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac12+\frac12\cos2\theta\right)d\theta\\ &=2r^2\left[\frac12\theta+\frac14\sin2\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\\ &=2r^2\left(\frac12\left(\frac{\pi}{2}\right)+\frac14\sin\left(2\left(\frac{\pi}{2}\right)\right)-\left(\frac12\left(-\frac{\pi}{2}\right)+\frac14\sin\left(2\left(-\frac{\pi}{2}\right)\right)\right)\right)\\ &=2r^2\left(\frac{\pi}{4}+\frac14\sin\pi+\frac{\pi}{4}+\frac14\sin\pi\right)\\ &=2r^2\left(\frac{\pi}{2}\right)\\ A&=\boxed{\pi r^2} \end{aligned}

Note by Gordon Chan
3 months, 2 weeks ago

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