Q. 265.0( 1 Vote )

Using integration

Answer :

y = x|x|


Now |x|, behaves differently when x < 0 and x > 0 but we are not concerned for x < 0 because we want to find the area in first quadrant


|x| = x for x > 0


Hence y = x2 for x > 0


x2 + y2 = 2 is a circle with center (0, 0) and radius √2


Let us find where does the parabola and circle intersect at


For this solve the equation of parabola and circle


Put y = x2 in equation of circle


x2 + x4 = 2


By observation x = 1 and x = -1 satisfies the equation but we want to find area in first quadrant hence x = 1


Now put x = 1 in parabola equation to get the y coordinate


y = 12


y = 1


Hence the parabola and circle intersect at (1, 1) in first quadrant


Roughly plot both the graphs y = x2 which is a parabola and the circle x2 + y2 = 2 and the area required is as shown



We have to find the shaded region OAB


Construct a line x = 1 and the area OAB will be the area under the circular arc BA minus the area under the parabola arc OA



First let us write the equations of circle and parabola in the from y = f(x)


Circle: x2 + y2 = 2



Parabola: y = x2


The area is given by


Area OAB = area under circle curve AB – area under parabola curve OA







The integral is of the form where a = √2


We know that


Hence






Hence area is square units


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

Draw a rough sketMathematics - Exemplar

Compute the area Mathematics - Exemplar

Using integrationMathematics - Board Papers

Find the area of Mathematics - Board Papers

Find the area of Mathematics - Exemplar

Evaluate <span laMathematics - Board Papers

The area of the rMathematics - Exemplar

Find the area bouMathematics - Exemplar