Question
Find the area enclosed by the circle $x^2 + y^2 = a^2$
✓
Answer
The equation of circle is $x^2 + y^2 = a^2 .... (i)$
Its centre is origin and radius $a.$
We know that circle $x^{2 }+ y^2 = a^2$ is symmetrical about both axes.
$\therefore$ required area = $4 \int \limits_{0}^{2} y d x=4 \int \limits_{0}^{2} \sqrt{a^{2}-x^{2}} d x$
$=4\left[\frac{x \sqrt{a^{2}-x^{2}}}{2}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right]_{0}^{a}$
$=4\left[\left(0+\frac{\mathrm{a}^{2}}{2} \sin ^{-1} 1\right)-\left(0+\frac{\mathrm{a}^{2}}{2} \sin ^{-1} 0\right)\right]$
$=4\left[\left(0+\frac{a^{2}}{2} \sin ^{-1} 1\right)-\left(0+\frac{a^{2}}{2} \sin ^{-1} 0\right)\right]$
$=4\left[\frac{a^{2}}{2} \times \frac{\pi}{2}\right]$ $\left[\because \sin ^{-1} 0=0\right]$
$=\pi a$
Its centre is origin and radius $a.$
We know that circle $x^{2 }+ y^2 = a^2$ is symmetrical about both axes.
$\therefore$ required area = $4 \int \limits_{0}^{2} y d x=4 \int \limits_{0}^{2} \sqrt{a^{2}-x^{2}} d x$
$=4\left[\frac{x \sqrt{a^{2}-x^{2}}}{2}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right]_{0}^{a}$
$=4\left[\left(0+\frac{\mathrm{a}^{2}}{2} \sin ^{-1} 1\right)-\left(0+\frac{\mathrm{a}^{2}}{2} \sin ^{-1} 0\right)\right]$
$=4\left[\left(0+\frac{a^{2}}{2} \sin ^{-1} 1\right)-\left(0+\frac{a^{2}}{2} \sin ^{-1} 0\right)\right]$
$=4\left[\frac{a^{2}}{2} \times \frac{\pi}{2}\right]$ $\left[\because \sin ^{-1} 0=0\right]$
$=\pi a$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
Evaluate the following integrals:
$\int\limits^4_0|\text{x}-1|\text{dx}$
→$\int\limits^4_0|\text{x}-1|\text{dx}$
$\int\frac{2\text{x}-1}{(\text{x}-1)^2}\text{dx}$
→Find which of the binary operations are commutative and which are associative.
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)
Using properties of determinants, prove that: $\begin{vmatrix}\sin\alpha&\cos\alpha&\cos(\alpha+\delta)\\\sin\beta&\cos\beta&\cos(\beta+\delta)\\\sin\gamma&\cos\gamma&\cos(\gamma+\delta)\end{vmatrix}=0$
→Differential equation $\text{x}\frac{\text{dy}}{\text{dx}}=1,\text{y}(1)=0$Function $\text{y}=\log\text{x}$
→Write the multiplication table for the set of integers modulo $5$.
→Compute the magnitude of the following vectors: $\vec a = \hat i + \hat j + \hat k,\vec b = 2\hat i - 7\hat j - 3\hat k,$ $\vec c = \frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j - \frac{1}{{\sqrt 3 }}\hat k$
→For what value of x is the matrix $\begin{bmatrix}6-\text{x}&4\\3-\text{x}&1\end{bmatrix}$ singular?
→Differentiate the function with respect to x : $\frac{{\sin \left( {ax + b} \right)}}{{\cos \left( {cx + d} \right)}}$
→