JEE Main & Advanced Mathematics Definite Integration Area of Bounded Regions

(1) The area bounded by a cartesian curve\[y=f(x)\], x-axis and ordinates \[x=a\] and \[x=b\] is given by     

Area \[=\int_{a}^{b}{y\,dx}=\int_{a}^{b}{f(x)dx}\]    

(2) If the curve \[y=f(x)\] lies below x-axis, then the area bounded by the curve \[y=f(x),\] the x-axis and the ordinates \[x=a\] and \[x=b\] is negative. So, area is given by \[\left| \int_{a}^{b}{y\,dx} \right|\].

(3) The area bounded by a cartesian curve\[x=f(y),\,\] y-axis and abscissae \[y=c\] and \[y=d\] is given by,

Area \[=\int_{c}^{d}{x\,dy=\int_{c}^{d}{f(y)dy}}\]

(4) If the equation of a curve is in parametric form, let \[x=f(t),\,\,y=g(t)\] then the area \[=\int_{a}^{b}{y\,dx}=\int_{{{t}_{1}}}^{{{t}_{2}}}{g(t)\,f'(t)\,dt}\] , where \[{{t}_{1}}\] and \[{{t}_{2}}\] are the values of \[t\]  respectively  corresponding to the values of \[a\] and \[b\] of \[x\].