Question
Determine the area under the curve $\text{y}=\sqrt{\text{a}^2-\text{x}^2}$ included between the lines x = 0 and x = a.
✓
Answer
We have $\text{y}=\sqrt{\text{a}^2-\text{x}^2}$ $\Rightarrow\ \text{y}^2=\text{a}^2-\text{x}^2$ $\Rightarrow\ \text{x}^2+\text{y}^2=\text{a}^2$ Graph of above function is a semi-circle lying above x-axis. The graph is as shown is the following figure.
From the figure, are of shaded region, $\text{A}=\int\limits^\text{a}_0\sqrt{\text{a}^2-\text{x}^2}\text{ dx}$ $=\bigg[\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\bigg]^\text{a}_0$ $=\bigg[0+\frac{\text{a}^2}{2}\sin^{-1}1-0\frac{\text{a}^2}{2}\sin^{-1}0\bigg]$ $=\frac{\text{a}^2}{2}\cdot\frac{\pi}{2}=\frac{\pi\text{a}^2}{4}\text{ sq. units}$
From the figure, are of shaded region, $\text{A}=\int\limits^\text{a}_0\sqrt{\text{a}^2-\text{x}^2}\text{ dx}$ $=\bigg[\frac{\text{x}}{2}\sqrt{\text{a}^2-\text{x}^2}+\frac{\text{a}^2}{2}\sin^{-1}\frac{\text{x}}{\text{a}}\bigg]^\text{a}_0$ $=\bigg[0+\frac{\text{a}^2}{2}\sin^{-1}1-0\frac{\text{a}^2}{2}\sin^{-1}0\bigg]$ $=\frac{\text{a}^2}{2}\cdot\frac{\pi}{2}=\frac{\pi\text{a}^2}{4}\text{ sq. units}$
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
find the area of the region enclosed by the parbola $x^2 = y$ and the line $y = x + 2$.
→If $\text{A}=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix},$ show that $A^2 - 7A + 10I_3 = 0$.
→Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Show that $\text{y}=\text{ae}^{2\text{x}}+\text{be}^{-\text{x}}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\frac{\text{dy}}{\text{dx}}-2\text{y}=0$
→Find the vector equation of the plane passing through points $3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},2\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}}$ and $7\hat{\text{i}}+6\hat{\text{k}}.$
→$\text{If (ax + b)} \text{e}^{\text{y/x}} = \text{x},\text{then show that}$
$\text{x}^{3} \bigg(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\bigg) = \bigg(\text{x}\frac{\text{dy}}{\text{dx}}- \text{y}\bigg)^{2} $
→$\text{x}^{3} \bigg(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\bigg) = \bigg(\text{x}\frac{\text{dy}}{\text{dx}}- \text{y}\bigg)^{2} $
Show that $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$ is decreases in the interval $[0,\infty)$ and increases in the interval $(-\infty,0].$
→To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of ₹ 20, ₹ 15 and ₹ 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.
By such exhibition, which values are generated in the students?
By such exhibition, which values are generated in the students?
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.
→'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 - 15x^2 + 36x + 1$
→$f(x) = 2x^3 - 15x^2 + 36x + 1$