MCQ
The area bounded by the parabolas $y=x^2$ and $y=1-x^2$ equals
- A$\frac{\sqrt{2}}{3}$
- ✓$\frac{2 \sqrt{2}}{3}$
- C$\frac{1}{3}$
- D$\frac{2}{3}$
✓
Answer
Correct option: B.
$\frac{2 \sqrt{2}}{3}$
b
(b)
(b)
We have, $y=x^2$ and $y=1-x^2$
Intersection point of $y=x^2$ and $y=1-x^2$ is $A\left(\frac{1}{\sqrt{2}}, \frac{1}{2}\right)$ and $C\left(\frac{-1}{\sqrt{2}}, \frac{1}{2}\right)$
Area of shaded region
$=2 \int \limits_0^{1 / \sqrt{2}}\left[\left(1-x^2\right)-\left(x^2\right)\right] d x$
$=2 \int \limits_0^{1 / \sqrt{2}}\left(1-2 x^2\right) d x$
$=2\left[x-\frac{2 x^3}{3}\right]_0^{1 / \sqrt{2}}$
$=2\left[\frac{1}{\sqrt{2}}-\frac{1}{3 \sqrt{2}}\right]$
$=\frac{2}{\sqrt{2}}\left[\frac{3-1}{3}\right]=\frac{2 \sqrt{2}}{3}$ 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
If $\left|\begin{array}{ll}3 & 3 \\ x & 1\end{array}\right|=\left|\begin{array}{cc}-3 & x \\ 1 & 1\end{array}\right|$ then value of $x$ is :
→Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{ PQ }=a \hat{ i }+b \hat{ j }$ and $\overrightarrow{ PS }=a \hat{ i }-b \hat{ j }$ are adjacent sides of $a$ parallelogram $P Q R S$. Let $\overrightarrow{ u }$ and $\overrightarrow{ v }$ be the projection vectors of $\overrightarrow{ w }=\hat{ i }+\hat{ j }$ along $\overrightarrow{ PQ }$ and $\overrightarrow{ PS }$, respectively. If $|\vec{u}|+|\vec{v}|=|\vec{w}|$ and if the area of the parallelogram $P Q R S$ is $8$ , then which of the following statements is/are $TRUE$?
→$(A)$ $a+b=4$
$(B)$ $a-b=2$
$(C)$ The length of the diagonal $P R$ of the parallelogram $P Q R S$ is $4$
$(D)$ $\overrightarrow{ w }$ is an angle bisector of the vectors $\overrightarrow{ PQ }$ and $\overrightarrow{ PS }$
If $A = \left[ {\begin{array}{*{20}{c}}{\cos \theta }&{ - \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right]$, then which of the following statements is not correct
→The least and greatest value of $f(x) = x^3 - 6x^2+ 9x$ in $[0, 6]$, are.
→For $I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$, if $I\left(\frac{\pi}{4}\right)=2^{1011}$, then
→A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20 \%$ and plant $T_2$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that
→$P$ (computer turns out to be defective given that it is produced in plant $T_1$ )
$=10 P\left(\right.$ computer turns out to be defective given that it is produced in plant $\left.T_2\right)$,
where $P(E)$ denotes the probability of an event $E$. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is
If $\text{AB}=\text{A}$ and $\text{BA = B}$ then $\text{B}^2 $ is equal to:
→Let $f(x) = max \,\{sin^{-1}x, cos^{-1}x\}$, then area bounded by $x = -1, x = 1, y = f(x)$ and $y = 0$ is -
→For linear programming $x+2 y \geq 10,3 x+4 y \leq 24$ and $x \geq 0, y \geq 0 \ldots \ldots \ldots .$ is not the corner point of feasible region.
→Let $\text{A}=\{\text{x}:-1\leq\text{x}\leq1\}$ and $f: A \rightarrow A$ such that $\text{f(x)}=\text{x}|\text{x}|,$ then $f$ is:
→