Area Between Two Curves





In this page area between two curves we are going to see example problems to understand how to find area between curves.

Area between two curves-Example

Example 1:

Find the area enclosed by the curves y² = x and y = x - 2

Solution:

First we need to draw the rough sketch for the given curves. For that let us find point of intersection of two curves. The first curve must be parabola and it is open right ward. Because we have square only for the variable y and it is positive. 

     y² = x

     y = √x    ---- (1)

     y = x - 2 ---- (2)

    (1)  =  (2)

       y = y

    √x = (x - 2)

 In order to remove square root we need to take square on both sides.

     (√x)² = (x - 2)²

           x = x² + 4 - 2(2) x

           x = x² + 4 - 4 x

           0 = x² + 4 - 4 x - x  

           0 = x² + 4 - 5x  

            x²  - 5x + 4  = 0  

           (x - 1) (x - 4) = 0

           x =1 and x = 4

Therefore the given parabola and line are intersecting at the points (1,0) and (4,0)


If we take the region about x-axis we need to split the region as two parts. For avoiding that let us take the region about y-axis.

So we need to take the limits as y = -1 and y = 2. We need to subtract the area below the parabola from the area below to the line. So that we will get the area of the shaded portion. Because we are taking the area about y-axis.

Equation of parabola : x = y²

Equation of the line : x = y + 2



   =   [((-1)²/2) + 2(-1) - ((-1)³/3) ] - [(2²/2) + 2(2) - (2³/3)]


   =   [(1/2) - 2 + (1/3) ] - [(4/2) + 4 - (8/3)]


   =   [(3 - 12 + 2)/6 ] - [2 + 4 - (8/3)]

   =   [-7/6 ] - [6 - (8/3)]

   =   [-7/6 ] - [(18 - 8)/3]

   =   [-7/6 ] - [10/3]

   =   [-7/6 ] - [20/6]

   =   -27/6 

   =   (9/2)  Square units.

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Area Between Two Curves to Area in Integration