Home
Class 12
MATHS
The area bounded by the curve xy^2 =a^...

The area bounded by the curve xy^2 =a^2 (a−x) and y-axis is

Text Solution

Verified by Experts

The correct Answer is:
`4pi` sq. units
We have curve `xy^(2)=4(2-x)`
`therefore" "x=(8)/(y^(2)+4)`
The given curve is symmetrical about x-axis and meets it at (2,0).
`"Also, when "yrarrpmoo,xrarr0.`
So, y-axis is asymptote to the curve.
The graph of the function is as shown in the following figure.

The line x=0, i.e., y-axis is asymptote.
Area of the shaded region `=2int_(0)^(oo)(8)/(y^(2)+4)dx`
`=16[(1)/(2)tan^(-1)""(y)/(2)]_(0)^(oo)=8xx(pi)/(2)`
`=4pi` sq. units
Promotional Banner

Topper's Solved these Questions

  • AREA

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 9.2|14 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 9.3|7 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Solved Examples|10 Videos

  • APPLICATIONS OF DERIVATIVES

    CENGAGE PUBLICATION|Exercise Subjective Type|2 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Comprehension|11 Videos

Similar Questions

Explore conceptually related problems

The area bounded by the curve y=x(1-log_(e)x) and x-axis is (a) (e^(2))/(4) (b) (e^(2))/(2) (c) (e^(2)-e)/(2) (d) (e^(2)-e)/(4)

Draw rough sketch of the area bounded by the curves x^(2) +y^(2) =2ax and y^(2) =ax and find its area.

If the area bounded by the curves y^2 = 4ax and x^2 = 4ay is (16a^2)/3 sq- unit then find the area bounded by y^2 = 2x and x^2 = 2y

Find the ratio in which the area bounded by the curves y^2=12 x and x^2=12 y is divided by the line x=3.

Mark the area bounded by the curve (y-1) (y-5) =4x and the y-axis and obtain the area by integration

The area bounded by the curve y = x^(3) , x-axis and the ordinates: x = - 1 and x= 1 is given by

The area enclosed by the curves xy^2=a^2(a-x) and (a-x)y^2=a^2x is

If the curve y= a sqrt(x) + bx passes through the point (1, 2) and the area bounded by the curve, the line x=4 and x-axis is 8 square unit, then the values of a and b are-

The area bounded by the curve y=3/|x| and y+|2-x|=2 is

Find the area bounded by the curve y=4x(x-1)(x-2) and the x-axis.