Question

Find the area of circle x^२ + y^२ =४ using integration​

Answer 1

Answer:

Equation of the circle is x

2

+y

2

=4

∴y=

4−x

2

But the area element is given by

∴ydx=

4−x

2

dx

Integrating both the sides with x ranging from -2 to 2

∴A=∫

−2

2

ydx=∫

−2

2

4−x

2

dx

Let x=2sinθ ⇒dx=2cosθdθ

∴A=∫

−π

π

4−4sin

2

θ

2cosθdθ=∫

−π

π

4cos

2

θdθ

=2∫

−π

π

(1+cos2θ)dθ

=4π

Answer 2

$\large\underline{\sf{Solution-}}$

Given equation of circle is

$\rm \: {x}^{2} + {y}^{2} = 4$

can be rewritten as

$\rm \: {(x - 0)}^{2} + {(y - 0)}^{2} = {2}^{2}$

It means, circle having centre (0, 0) and radius, r = 2 units.

Now, see the attachment.

We concluded that circle divided in to 4 equal parts in 4 quadrants, So we find the area in first quadrant with respect to x - axis from x = 0 to x = 2 and multiply by 4 to find the required area.

So, required area of circle is

$\rm \: = \:4 \: \displaystyle\int_{0}^{2}\rm y \: dx$

$\rm \: = \:4 \: \displaystyle\int_{0}^{2}\rm \sqrt{4 - {x}^{2} } \: dx$

$\rm \: = \:4 \: \displaystyle\int_{0}^{2}\rm \sqrt{ {2}^{2} - {x}^{2} } \: dx$

$\rm \: = \:4 \bigg |\dfrac{x}{2} \sqrt{ {2}^{2} - {x}^{2}} + \dfrac{ {2}^{2} }{2} {sin}^{ - 1} \dfrac{x}{2} \bigg| _{0}^{2}$

$\rm \: = \:4 \bigg |\dfrac{x}{2} \sqrt{ 4 - {x}^{2}} + 2{sin}^{ - 1} \dfrac{x}{2} \bigg| _{0}^{2}$

$\rm \: = \:4\bigg( \dfrac{2}{2} \sqrt{ 4 - 4} + 2{sin}^{ - 1} \dfrac{2}{2} - \dfrac{0}{2} \sqrt{ 4 - {0}^{2}} - 2{sin}^{ - 1} \dfrac{0}{2}\bigg)$

$\rm \: = \:4 \times 2{sin}^{ - 1} 1$

$\rm \: = \: 8 \times \dfrac{\pi}{2}$

$\rm \: = \: 4\pi \: square \: units$

$\rule{190pt}{2pt}$

Formulae Used :-

$\boxed{ \rm{ \:\displaystyle\int\rm \sqrt{ {a}^{2} - {x}^{2} }dx = \dfrac{x}{2} \sqrt{ {a}^{2} - {x}^{2}} + \frac{ {a}^{2} }{2} {sin}^{ - 1} \dfrac{x}{a} + c\: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$