To find the area of the region defined by $\{(x, y) : x^2 \le y \le x\}$, we need to calculate the definite integral of the difference between the upper curve ($y=x$) and the lower curve ($y=x^2$) between their points of intersection.
Finding Intersection Points
Set the equations equal to find the intersection points:
$x^2 = x$
$x^2 - x = 0$
$x(x - 1) = 0$
The points of intersection are at $x=0$ and $x=1$.
Setting Up the Integral
In the interval $[0, 1]$, the line $y=x$ is above the parabola $y=x^2$. We can verify this by testing a point, e.g., $x=0.5$: $y=0.5$ for the line and $y=(0.5)^2=0.25$ for the parabola. Since $0.5 > 0.25$, $y=x$ is the upper boundary.
The area $A$ is given by the integral:
$A = \int_{0}^{1} (\text{upper curve} - \text{lower curve}) \, dx$
$A = \int_{0}^{1} (x - x^2) \, dx$
Calculating the Area
Evaluate the definite integral:
$A = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{1}$
$A = \left( \frac{1^2}{2} - \frac{1^3}{3} \right) - \left( \frac{0^2}{2} - \frac{0^3}{3} \right)$
$A = \left( \frac{1}{2} - \frac{1}{3} \right) - (0 - 0)$
$A = \frac{3 - 2}{6}$
$A = \frac{1}{6}$
The area of the region is $\frac{1}{6}$ square units.
Important Questions from Miscellaneous
- Which of the following relationships is/are not true?
(A). Most probable velocity = $\sqrt{\frac{2RT}{M}}$
(B). PV = $\frac{3}{2}kT$
(C). Compressibility factor Z = $\frac{pV}{nRT}$
(D). Average kinetic energy of gas = $\frac{1}{2}kT$
Choose the correct answer from the options given below - Match List-I with List-II
List-I List-II Electronic Configuration First Ionisation energy (kJ mol$^{-1}$) (A). ns$^2$ (I). 2100 (B). ns$^2$np$^1$ (II). 1400 (C). ns$^2$np$^3$ (III). 800 (D). ns$^2$np$^6$ (IV). 900
Choose the correct answer from the options given below: - The shielding constant of a 2p electron (calculated using Slater's rules) is
- Match List-I with List-II
List-I List-II Spectroscopy Property (A). Raman (I). Polarizability (B). FTIR (II). Dipole Moment (C). UV-Visible (III). Absorbance (D). NMR (IV). Spin
Choose the correct answer from the options given below: - The structure of protein comprises of:
(A). Primary structure of protein is associated with amino acids
(B). Secondary structure of protein is associated to peptides
(C). Tertiary structure of protein is associated with polypeptide chains
(D). Quaternary structure of protein is associated with polypeptide chains
Choose the correct answer from the options given below:
