PURE MATHEMATICS - Integration

 

Area under a Curve

 

 

different axes

positive and negative area

area between 2 curves

 

 

Area Under a Curve, related to different axes

 

 

curve area

 

 

     

 

 

Example

Find the area 'A' enclosed by the x-axis, x = 2 , x = 4 and the graph of

 

 

curve area problem #1

 

 

 

         

 

                         

 

                  

 

           

 

 

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Positive and Negative Area

 

 

curve area #2

 

 

                      

 

 

NB: This expression calculates the absolute area between the curve the vertical lines at 'a' and 'b' and the x-axis. It takes no account of sign.

 

If sign were an issue then the two integrals on the first line would be added and not subtracted.

 

Unless told differently, assume that the 'absolute area' is required.

 

 

Example

 

Find the area 'A' enclosed by x-axis, x = 1 , x = 8 and the graph .

 

 

area under curve problem#2

 

 

The curve crosses the x-axis at y = 0.

 

therefore,                                    

 

 

Sine is zero when the angle is 0, 180 or 360 deg. (ie at zero, pi and 2 pi)

 

 

 

 

                                                                                  

 

              

 

                  

 

                  

 

                             

 

                                      

 

 

 

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Area Bounded by Two Curves

 

 

curve area #3

 

 

 

 

Example

 

To 3 d.p. calculate the area 'A' included between the curves y = x2/2 and y = (1.75)x .

 

 

First find the x-value where the curves of each function cross.

 

               

where the curves cross

 

 

 

  ,     therefore 

 

 

 

included area problem

 

 

 

The area 'A' is the difference between the area under the straight line and the area under the parabola, from x=0 to x=3.5 .

 

 

         

 

 

 

 

               

                                   

 

 

 

 

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