Find the area enclosed by the parabola 4y=3x2 and the line 2y=3x+12
Find the area enclosed by the parabola 4y=3x2 and the line 2y=3x+12
Solution and Explanation
The area enclosed between the parabola,4y=3x2,and the line,2y=3x+12,is
represented by the shaded area OBAO as
The points of intersection of the given curves are A(–2,3)and(4,12).
We draw AC and BD perpendicular to x-axis.
∴Area OBAO=Area CDBA–(Area ODBO+Area OACO)
=
\[\int_{-2}^{1} \frac12(3x+12) \,dx\]\[-\int_{-2}^{1} \frac{3x^2}{4} \,dx\]=\(\frac12\)[\(\frac{3x^2}{2}\)+12x]4-2-\(\frac34\)[x3/3]4-2
=\(\frac12\)[24+48-6+24]-\(\frac14\)[64+8]
=\(\frac12\)[90]-\(\frac14\)72]
=45-18
=27units.
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII applications of integrals Questions
- Find the area of the circle 4x2+4y2=9 which is interior to the parabola x2=4y
- Find the area bounded by curves(x-1)2+y2=1 and x2+y2=1
- Area bounded by the curve y=x3, the x-axis and the ordinates x=2 and x=1 is
Using integration finds the area of the region bounded by the triangle whose vertices are (–1, 0),(1, 3)and(3, 2).
Using integration find the area of the triangular region whose sides have the equations y =2x+1,y=3x+1 and x=4.
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is