Multiple Integrals, Areas & Volumes

1. The area bounded by the curve x = f(y), the x-axis and the

        abscissa y = a, y = b is

     Length of Curves:

1. The length of the arc of the curve y = f(x) between the points where x = a and x = b is
2. The length of the arc of the curve x = f(y) between the point where y =a and y = b, is 
    

1. The length of the arc of the curve between the points where t =a and t =b, is

2. The length of the arc of the curve between the point where and θ = α and θ =β, is

3. The length of the arc of the curve  between the point where r = a and r = b, is  

   Change of order of integration:

   In a double integral with variable limits, the change of order of integration changes the limit of integration. While doing so, sometimes it is required to split up the region of integration, and the given integral is expressed as the sum of a number of double integrals with changed limits. 

   The change of order of integration quite often facilities the evaluation of a double integral.