Find the area of the region bounded by the parabola y=x2 and y=|x|
Find the area of the region bounded by the parabola y=x2 and y=|x|
Solution and Explanation
The area bounded by the parabola,x2=y and the line,y=|x|,can be represented as
The given area is symmetrical about y-axis.
∴Area OACO=Area ODBO
The point of intersection of parabola,x2=y,and line,y=x,is A(1,1).
Area of ΔOAB=1/2×OB×AB=1/2×1×1=1/2
Area of OBACO=∫10ydx=∫10x2dx=[x3/3]10=1/3
⇒Area of OACO=Area of ΔOAB-Area of OBACO
=1/2-1/3
=1/6
Therefore,required area=2[1/6]=1/3units.
Top Questions on Conic sections
- Let the ellipse \[ E:\ \frac{x^2}{144}+\frac{y^2}{169}=1 \] and the hyperbola \[ H:\ \frac{x^2}{16}-\frac{y^2}{2^2}=1 \] have the same foci. If \(e\) and \(L\) respectively denote the eccentricity and the length of the latus rectum of \(H\), then the value of \(24(e+L)\) is:
- JEE Main - 2026
- Mathematics
- Conic sections
- Let \(P_1 : y = 4x^2\) and \(P_2 : y = x^2 + 27\) be two parabolas. If the area of the bounded region enclosed between \(P_1\) and \(P_2\) is six times the area of the bounded region enclosed between the line \(y = x\), the line \(x = 0\), and \(P_1\), then the required value is:
- JEE Main - 2026
- Mathematics
- Conic sections
- An ellipse has its centre at \((1,-2)\), one focus at \((3,-2)\) and one vertex at \((5,-2)\). Then the length of its latus rectum is:
- JEE Main - 2026
- Mathematics
- Conic sections
- Let the foci of a hyperbola coincide with the foci of the ellipse \(\frac{x^2}{36} + \frac{y^2}{16} = 1\). If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:
- JEE Main - 2026
- Mathematics
- Conic sections
- Let \( y^2 = 12x \) be the parabola with its vertex at \( O \). Let \( P \) be a point on the parabola and \( A \) be a point on the \( x \)-axis such that \( \angle OPA = 90^\circ \). Then the locus of the centroid of such triangles \( OPA \) is :
- JEE Main - 2026
- Mathematics
- Conic sections
Questions Asked in CBSE CLASS XII exam
- Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]
- CBSE CLASS XII - 2026
- Differential Equations
- A square loop of side 0.50 m is placed in a uniform magnetic field of 0.4 T perpendicular to the plane of the loop. The loop is rotated through an angle of 60° in 0.2 s. The value of emf induced in the loop will be:
- CBSE CLASS XII - 2026
- Electromagnetic induction
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
- CBSE CLASS XII - 2026
- Integration
- Consider a cylindrical conductor of length \( l \) and area of cross-section \( A \). Current \( I \) is maintained in the conductor and electrons drift with velocity \( \vec{v}_d \, (|\vec{v}_d| = \frac{eE}{m} \tau) \), where symbols have their usual meanings. Show that the conductivity of the material of the conductor is given by \[ \sigma = \frac{n e^2 \tau}{m}. \]
- CBSE CLASS XII - 2026
- Current Density
- The painting Hazrat Nizamuddin Auliya and Amir Khusro belongs to which sub-school of Deccan Miniature?
- CBSE CLASS XII - 2026
- Indian Miniature Paintings and Museums