Applications of Integrals - GATE Study 
                                                    Material in PDF 
                                                                         
                  In the previous article we have seen some introduction about integrals. In these free 
                  GATE 2018 Notes we will discuss about the Applications of Integrals. Generally, 
                  integrations are used to find the areas, volumes, and length of curves.   
                  This study material is useful for GATE CS, GATE CE, GATE ME, GATE EC and GATE 
                  EE. Also useful for recruitment exams like BARC, DRDO, IES, ISRO, BSNL etc. 
                  These Notes can be downloaded in PDF so that your GATE preparation is made easy 
                  and you ace your paper. Before you go ahead though, make sure to read the basics in 
                  Engineering Mechanics.  
                  Recommended Reading – 
                                                         Types of Matrices 
                                                     Properties of Matrices 
                                            Rank of a Matrix & Its Properties 
                                     Solution of a System of Linear Equations 
                                               Eigen Values & Eigen Vectors 
                                              Linear Algebra Revision Test 1 
                                                       Laplace Transforms 
                  1 | P a g e           Limits, Continuity & Differentiability 
                                                                                                                              
                                                         Mean Value Theorems                                                            
                                                                Differentiation 
                                                         Partial Differentiation 
                                                          Maxima and Minima 
                                     Methods of Integration & Standard Integrals 
                                                                              
                   Here are some of the most important Applications of Integrals. 
                   1. Area Calculation 
                   2. Length of Curve Calculation 
                   3. Volume Calculation 
                   1. Area Calculation Using Integration  
                   If the curve is symmetrical about x − axis then area  =                   by dx  
                                                                                           ∫
                                                                                            a
                   If the curve is symmetrical about y − axis then area  =                   bx dy  
                                                                                           ∫
                                                                                            a
                   Area enclosed between two curves let y  = f(x),  y  = g(x) then   
                                                                        1            2
                                b
                               ∫ (y −y )dx            if y  >y  
                   Area =   a        1     2              1      2   
                                 b
                                ∫ (y −y )dx            if y  >y
                                 a    2     1              2      1.
                    
                   Example 1:   
                                                                                  2
                   The area bounded between the parabolas y = x  and the straight line y = x is   
                   Solution:  
                   2 | P a g e  
                                                                                                                                        
                                                                                                 
                                                                           
                       1      2
              Area  = ∫ (x−x )dx  
                       0
                  2   3 1
              =[x −x ]   
                 2   3 0
                1  1   1
              = − =   
                2  3   6
               
              Example 2:  
                                              2           2
              Find the area bounded between y  = 4x  and x  = 4y is   
              Solution:  
              3 | P a g e  
                                                                                                 
                                                                                                        
                                                                               
                 A =  4( 4x−x2)⋅dx  
                    ∫ √
                     0         4
                      3 4
                     x2      1  3 4
               =2⋅[3] − [x ]0  
                            12
                      2 0
                 4      4
                  [ ]     [ ]
               = 8 − 4  
                 3      3
                 4       16
               = ×4=   
                 3        3
                
               Note:   
               The area enclosed between  y2 = 4ax and x2 = 4by is 16ab.  
                                                                       3
                
                
               2. Length of the Curve Calculation using Integration  
               4 | P a g e