Let $A$ be the area bounded by the curve $y=x|x-3|$, the $x$-axis and the ordinates $x=-1$ and $x=2$ Then $12 A$ is equal to _____
Correct Answer: 62
Solution and Explanation
Computing the area: \[ A = \int_{-1}^{0} (x^2 - 3x) dx + \int_{0}^{2} (3x - x^2) dx \] \[ A = \left[ \frac{x^3}{3} - \frac{3x^2}{2} \right]_{-1}^{0} + \left[ \frac{3x^2}{2} - \frac{x^3}{3} \right]_{0}^{2} \] \[ A = \frac{11}{6} + \frac{10}{3} - \frac{31}{6} \] \[ \Rightarrow 12A = 62 \]
Top Questions on Area under Simple Curves
- Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta\pi + \gamma \); \( \beta, \gamma \) are integers, then the value of \( |\beta - \gamma| \) equals:
- JEE Main - 2025
- Mathematics
- Area under Simple Curves
If the area of the region \[ \{(x, y) : |4 - x^2| \leq y \leq x^2, y \leq 4, x \geq 0\} \] is \( \frac{80\sqrt{2}}{\alpha - \beta} \), where \( \alpha, \beta \in \mathbb{N} \), then \( \alpha + \beta \) is equal to:
- JEE Main - 2025
- Mathematics
- Area under Simple Curves
- The area of the region, inside the circle \((x-2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is:
- JEE Main - 2025
- Mathematics
- Area under Simple Curves
- What is the area of a triangle with base 8 cm and height 6 cm?
- VITEEE - 2025
- Mathematics
- Area under Simple Curves
The area of the region enclosed between the curve \( y = |x| \), x-axis, \( x = -2 \)} and \( x = 2 \) is:
- CBSE CLASS XII - 2025
- Mathematics
- Area under Simple Curves
Questions Asked in JEE Main exam
- Let circle $C$ be the image of
$$ x^2 + y^2 - 2x + 4y - 4 = 0 $$
in the line
$$ 2x - 3y + 5 = 0 $$
and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre $O$ of $C$.
If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $\frac{1}{6}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to: - Assume a living cell with 0.9%(\(w/w\)) of glucose solution (aqueous). This cell is immersed in another solution having equal mole fraction of glucose and water. (Consider the data up to first decimal place only) The cell will:
- JEE Main - 2025
- osmosis
Given below are two statements:
Statement (I):
are isomeric compounds.
Statement (II):
are functional group isomers.
In the light of the above statements, choose the correct answer from the options given below:- JEE Main - 2025
- Isomerism
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
The effect of temperature on the spontaneity of reactions are represented as: Which of the following is correct?
- JEE Main - 2025
- Thermodynamics and Work Done
Concepts Used:
Applications of Integrals
There are distinct applications of integrals, out of which some are as follows:
In Maths
Integrals are used to find:
- The center of mass (centroid) of an area having curved sides
- The area between two curves and the area under a curve
- The curve's average value
In Physics
Integrals are used to find:
- Centre of gravity
- Mass and momentum of inertia of vehicles, satellites, and a tower
- The center of mass
- The velocity and the trajectory of a satellite at the time of placing it in orbit
- Thrust