Consider the line x= 3 y and the circle x^2+y^2=4 . What is the area of the region in the first quadrant enclosed by the x -axis, the line x= 3 y and the circle?

Step-by-step solution

aligned & x^2+y^2=4 and x= 3 y & P( 3 , 1)= First quadrant aligned The point of intersection of the line and the circle in the first quadrant is ( 3 , 1) . Area of OPA = Area of OPB + Area of PAB = 1 2 OB PB = 1 2 3 1= 3 2 aligned & Area of PAB _ 3 ^2 ydx & = _ 3 ^2 4-x^2 d x= [ x 2 4-x^2 + 4 2 ⁻¹ x 2 ]_ 3 ^2 & [2 2 - 3 2 [ 4-3 ]-2 ⁻¹ 3 2 ] & = - 3 2 - 2 3 = 3 - 3 2 aligned Area enclosed by x -axis, the line = 3 . y , and the circle x^2+y^2=4 in the first quadrant = 3 2 + 3 - 3 2 = 3