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

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

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Answer 1 · 2014-09-08T03:13:11

By using polar coordinates, the area of a circle centered at the origin with radius $R$ $R$ can be expressed:
$A = \int_{0}^{2 π} \int_{0}^{R} r d r d θ = π R^{2}$ $A=int_0^{2pi}int_0^R rdrd theta=piR^2$

Let us evaluate the integral,
$A = \int_{0}^{2 π} \int_{0}^{R} r d r d θ$ $A=int_0^{2pi}int_0^R rdrd theta$
by evaluating the inner integral,
$= \int_{0}^{2 π} {[\frac{r^{2}}{2}]}_{0}^{R} d θ = \int_{0}^{2 π} \frac{R^{2}}{2} d θ$ $=int_0^{2pi}[{r^2}/2]_0^R d theta=int_0^{2pi}R^2/2 d theta$
by kicking the constant $\frac{R^{2}}{2}$ $R^2/2$ out of the integral,
$\frac{R^{2}}{2} \int_{0}^{2 π} d θ = \frac{R^{2}}{2} {[θ]}_{0}^{2 π} = \frac{R^{2}}{2} \cdot 2 π = π R^{2}$ $R^2/2int_0^{2pi} d theta=R^2/2[theta]_0^{2pi}=R^2/2 cdot 2pi=piR^2$

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

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