Applications Of Integrals

The intersection point of parabolas

y%3Dx%5E%7B2%7D

and

x%3Dy%5E%7B2%7D

y%3D%5Cleft(y%5E%7B2%7D%5Cright)%5E%7B2%7D

%5CRightarrow%20y%3Dy%5E%7B4%7D

%5CRightarrow%20y%3D0%2C%20y%3D1

%5CRightarrow%20x%3D0%2C%20x%3D1

So, the intersection point in

O(0%2C0)

and

(1%2C1)

%5Ctherefore

Required area

%3D%5Cint_%5Climits%7B0%7D%5E%7B1%7D%5Cleft(y_%7B2%7D-y_%7B1%7D%5Cright)%20d%20x

%3D%5Cint_%5Climits%7B0%7D%5E%7B1%7D%5Cleft(%5Csqrt%7Bx%7D-x%5E%7B2%7D%5Cright)%20d%20x

%3D%5Cleft%5B%5Cfrac%7Bx%5E%7B3%20%2F%202%7D%7D%7B3%20%2F%202%7D-%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%5Cright%5D_%7B0%7D%5E%7B1%7D%3D%5Cleft%5B%5Cfrac%7B2%7D%7B3%7D%20x%5E%7B3%20%2F%202%7D-%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%5Cright%5D_%7B0%7D%5E%7B1%7D

%3D%5Cleft%5B%5Cfrac%7B2%7D%7B3%7D(1)-%5Cfrac%7B1%7D%7B3%7D(1)%2B0-0%5Cright%5D

%3D%5Cleft%5B%5Cfrac%7B2%7D%7B3%7D-%5Cfrac%7B1%7D%7B3%7D%5Cright%5D%3D%5Cfrac%7B1%7D%7B3%7D

We know that, if parabolas are

y%5E%7B2%7D%3D4%20a%20x

and

x%5E%7B2%7D%3D4by

, then area of bounded region is

%5Cfrac%7B4%20a%20%5Ccdot%204%20b%7D%7B3%7D.

%5Ctherefore

Required area

%3D%5Cfrac%7B1%20%5Ctimes%201%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D

Chapter Questions
2 MCQs