calculating area using integration

calculation of area using integration (calculus method)
The area of the region bounded by y = f(x) , the x-axis , the horizontal lines x = a , x = b is given by

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The area of the region bounded by x = g(y) , the y-axis , the vertical lines y = c , y = d is given by
some problems on finding area using integration

area under one arch of the cycloid x = a(t-sint) , y =a(1-cost) ----------problem on area by integration

area between y²= x and x²=y -------------area between y²= x and x²=y

find the area of the region {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) more explanation on this area question

area bounded by y = x ² +2 and y = 3x ----------area between two curves
area of the cardioid r = a(1 - cosθ ) in polar form ----------- area in polar form