JEE Main & Advanced Mathematics Definite Integration Area Between Two Curves

(1) When both curves intersect at two points and their common area lies between these points: If the curves \[{{y}_{1}}={{f}_{1}}(x)\] and \[{{y}_{2}}={{f}_{2}}(x),\] where\[\,{{f}_{1}}(x)\,>\,{{f}_{2}}(x)\] intersect in two points \[A(x=a)\] and \[B(x=b)\], then common area between the curves is  \[=\int\limits_{a}^{b}{({{y}_{1}}-{{y}_{2}})\,dx}\]\[=\int\limits_{a}^{b}{[{{f}_{1}}(x)-{{f}_{2}}(x)]\,dx}\].

(2) When two curves intersect at a point and the area between them is bounded by x-axis: Area bounded by the curves \[{{y}_{{}}}={{f}_{1}}(x),{{y}_{2}}={{f}_{2}}(x)\,\,\text{and}\,x-\text{axis}\]is  \[\int\limits_{a}^{\alpha }{{{f}_{1}}(x)dx+\int\limits_{\alpha }^{b}{{{f}_{2}}(x)dx}}\],

where \[P(\alpha ,\beta )\,\]is the point of intersection of the two curves.

(3) Positive and negative area : Area is always taken as positive. If some part of the area lies above the x-axis and some part lies below x-axis, then the area of two parts should be calculated separately and then add their numerical values to get the desired area.