Question
Using definite intergeals, find the area of the circle $x^2+ y^2 = a^2$.
✓
Answer
Area of the circle $x^2+ y^2 = a^2$ will be thr $4$ times the area enclosed between $x = 0$ and $x = a$ in the first quadrant which is shaded.
$\text{A}=4\int\limits_{0}^{a}|\text{y}|\text{dx} $
$=4\int\limits_{0}^{a}\Big(\sqrt{\text{a}^{2}-\text{x}^{2}}\Big)\text{dx}$
$=4\Big[\frac{1}{2}\text{x}\sqrt{\text{a}^{2}-\text{x}^{2}}+\frac{1}{2}\text{a}^{2}\sin^{-1}\frac{\text{x}}{\text{a}}\Big]^{\text{a}}_{0}$
$=4\big[0+\frac{1}{2}\text{a}^{2}\sin^{-1}1\big] $
$=4\Big[\frac{1}{2}\text{a}^{2}\frac{\pi}{2}\Big] $
$=\text{a}^{2}\pi\ \text{sq.}\ \text{units}$
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
Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
→$\int\limits^{\frac{\pi}{3}}_\frac{\pi}{6}\frac{1}{1+\cot^{\frac{3}{2}}\text{x}}\text{ dx}$
Solve the following systems of linear equations by cramer's rule:
2x - 3z + w = 1,
x - y + 2w = 1,
-3y + z + w = 1,
x + y + z = 1
Find non-zero values of x satisfying the matrix equation:$\text{x}\begin{bmatrix}2\text{x}&2\\3&\text{x}\end{bmatrix}+2\begin{bmatrix}8&5\text{x}\\4&4\text{x}\end{bmatrix}=2\begin{bmatrix}(\text{x}^2+8)&24(10)&6\text{x}\end{bmatrix}$
→Evaluate the following definite integral as limit of sum:
$\int\limits_{1}^{4}(\text{x}^{2}-\text{x)}\ \text{dx}$
→$\int\limits_{1}^{4}(\text{x}^{2}-\text{x)}\ \text{dx}$
Find the equation of the tangents to the curve $3x^2 - y^2 = 8$, which passes through the point $\big(\frac{4}{3},0\big)$
→Prove that the function f given by $\text{f}(\text{x})=\log\cos\text{x}$ is strictly increasing on $\Big(-\frac{\pi}{2},0\Big)$ and strictly decreasing on $\Big(0,\frac{\pi}{2}\Big).$
→Let L be the set of all lines in xy plane and R be the relation in L. define as $s R =\left\{\left( L _1, L_2\right): L _1 \| L _2\right\}$ Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4
→A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of A and ₹ 80 on each piece of type ₹ 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
Evaluate the following integrals as limit of sum:
$\int\limits^3_1(3\text{x}-2)\text{dx}$
→$\int\limits^3_1(3\text{x}-2)\text{dx}$