The area bounded by the curve
\(y=|x^2−9| \)
and the line y = 3 is
The area bounded by the curve
\(y=|x^2−9| \)
and the line y = 3 is
\(4(2\sqrt3+\sqrt6-4)\)
\(4(4\sqrt3+\sqrt6-4)\)
\(8(4\sqrt3+3\sqrt6-9)\)
\(8(4\sqrt3+\sqrt6-9)\)
The Correct Option is A
Solution and Explanation
The correct asnwer is (D) : \(8(4\sqrt3+\sqrt6-9)\)
\(=2\int^{3}_{0}(\sqrt{9+y}-\sqrt{9-y})dy + 2\int^{9}_{3}(\sqrt{9-y})dy\)
\(=2\int^{3}_{0}((9+y)^{1/2}-(9-y)dy + \int^{9}_{3}({9-y})^{1/2})dy\)
\(=2[\frac{2}{3}[(9+y)^{3/2}]^{3}_{0} + \frac{2}{3}[(9-y)^{3/2}]^{3}_{0}-\frac{2}{3}[(9-y)^{3/2}]^{3}_{0}]\)
\(= \frac{4}{3}[12\sqrt{12}-27+6\sqrt6-27-(0-6\sqrt6)]\)
\(=\frac{4}{3}[24\sqrt3+12\sqrt6-54]\)
\(=8(4\sqrt3+2\sqrt6-9)\)
Top Questions on Area between Two Curves
- Let $A_1$ be the bounded area enclosed by the curves $y=x^2+2$, $x+y=8$ and $y$-axis that lies in the first quadrant. Let $A_2$ be the bounded area enclosed by the curves $y=x^2+2$, $y^2=x$, $x=2$ and $y$-axis that lies in the first quadrant. Then $A_1-A_2$ is equal to
- JEE Main - 2026
- Mathematics
- Area between Two Curves
- The area (in sq. units) of the region bounded by the curves \( y = x^2 \) and \( y = 8 - x^2 \) is
- AP EAPCET - 2025
- Mathematics
- Area between Two Curves
The eccentricity of the curve represented by $ x = 3 (\cos t + \sin t) $, $ y = 4 (\cos t - \sin t) $ is:
- MHT CET - 2025
- Mathematics
- Area between Two Curves
- If the area of the region $ \{(x, y) : 1 + x^2 \leq y \leq \min(x + 7, 11 - 3x)\} $ is $ A $, then $ 3A $ is equal to:
- JEE Main - 2025
- Mathematics
- Area between Two Curves
If the area of the region $\{ (x, y) : |x - 5| \leq y \leq 4\sqrt{x} \}$ is $A$, then $3A$ is equal to
- JEE Main - 2025
- Mathematics
- Area between Two Curves
Questions Asked in JEE Main exam
MX is a sparingly soluble salt that follows the given solubility equilibrium at 298 K.
MX(s) $\rightleftharpoons M^{+(aq) }+ X^{-}(aq)$; $K_{sp} = 10^{-10}$
If the standard reduction potential for $M^{+}(aq) + e^{-} \rightarrow M(s)$ is $(E^{\circ}_{M^{+}/M}) = 0.79$ V, then the value of the standard reduction potential for the metal/metal insoluble salt electrode $E^{\circ}_{X^{-}/MX(s)/M}$ is ____________ mV. (nearest integer)
[Given : $\frac{2.303 RT}{F} = 0.059$ V]- JEE Main - 2026
- Electrochemistry
An infinitely long straight wire carrying current $I$ is bent in a planar shape as shown in the diagram. The radius of the circular part is $r$. The magnetic field at the centre $O$ of the circular loop is :
- JEE Main - 2026
- Current electricity
- For two identical cells each having emf \(E\) and internal resistance \(r\), the current through an external resistor of \(6\,\Omega\) is the same when the cells are connected in series as well as in parallel. The value of the internal resistance \(r\) is ________ \(\Omega\).
- JEE Main - 2026
- Current electricity
- The half-life of $^{65}$Zn is 245 days. After $x$ days, 75% of the original activity remained. The value of $x$ in days is __________ (Nearest integer).}
(Given: $\log 3 = 0.4771$ and $\log 2 = 0.3010$)- JEE Main - 2026
- Surface Chemistry
- The hydrogen spectrum consists of several spectral lines in Lyman series (L$_1$, L$_2$, L$_3$ ...; L$_1$ has lowest energy among Lyman series). Similarly, it consists of several spectral lines in Balmer series (B$_1$, B$_2$, B$_3$ ...; B$_1$ has lowest energy among Balmer lines). The energy of L$_1$ is $x$ times the energy of B$_1$. The value of $x$ is ___ $\times 10^{-1}$. (Nearest integer)
- JEE Main - 2026
- Atomic Structure
Concepts Used:
Area between Two Curves
Integral calculus is the method that can be used to calculate the area between two curves that fall in between two intersecting curves. Similarly, we can use integration to find the area under two curves where we know the equation of two curves and their intersection points. In the given image, we have two functions f(x) and g(x) where we need to find the area between these two curves given in the shaded portion.
Area Between Two Curves With Respect to Y is
If f(y) and g(y) are continuous on [c, d] and g(y) < f(y) for all y in [c, d], then,
