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.