1)  Numerical Integration deals with areas of closed regions.
2) Closed regions are formed between two intersections curves; between a curve and a straight line; a curve and the coordinate axes, a line in intercept form and the coordinate axes and a curve represented by a quadratic equation and either x or y-axis.
3) For measuring the areas, the following guidelines are to be noted:
      (i) Know the nature of the curve.
      (ii) Obtain the points of intersection as under the question.
      (iii) Evaluate ∫ y dx or ∫ x dy by using the appropriate limits of integration.
      (iv) The process of definite integration yields the required area.
      (v) Shade the region for which the area is to be calculated.
4) Prominent Areas:
(i)  Area of circle  x²  +  y²  =  r²  is  Π r².
(ii)   Area of ellipse     is Πab.

(iii)  Area between the parabola   y²  =  4ax  and   x²  =  4by is    ab.
(iv)  Area between the parabola  y²   =  4 ax and  x² = 4ay is   a² .
(v)   Area bounded by  y²  =  4ax and   y  =  mx is  .
(vi)   Area bounded by  y²  =  4ax with  x  =  a is .
(vii)  Area bounded by  y²  =  4bx with  y  =  b is  .
(viii) Area bounded by  x²  =  4by with  x  =  a  and  x - axis  is    .
(ix)  Area bounded by  x²  =  4ay  with  x  =  a  and  x - axis  is 

  .
(x)   Area bounded by  y  =  sin ax  with  x - axis in the interval [ 0,  Π ]  is  .
(xi)  Area bounded by  y  =  sin ax  with  x - axis in the interval  [ 0,  2Π ]  is  .
(xii)  Area bounded by  y  =  ax²  +  bx  +  c  with  x - axis  is   .

(xiii)   Area bounded by  x  =  ay²  +  by  +  c  with y - axis  is 
(xiv)   Area enclosed by    is 2.