integral - area inside circle above (root3)x?

Question

i'm studying for my midterm by doing old exams but can't get this question, so i'd appreciate anyone's help very much

find area inside circle x^2+y^2=4, and above the line y=(square root of 3)x

i tried finding the intersections, and i think they are at (1, root3) and (-1, -root3). then i got confused as to how to split up the integral into two sections. thanks in advance for your help.

Yeah, I also realized that the circle is just divided in half, but the question asks for an integral so i can't just cheat like that :P

And I just feel like when the line splits through the circle is when I need to separate it into two integrals, perhaps one from [-2,-1] and another from [-1,2] ?

Answer 1

Wouldn't that just divide the circle in half? You have a circle of radius 2... so half the area would be 2pi?

Answer 2

I guess if you are doing calculus you want the integral, but I would get this with simple geometry: the sector is 120 degrees because it is a 30-60-90 triangle, so 4pi/3 - sqrt(3).

Why do you need to split the integral up? integral of sqrt(4 - x^2) integrates easily.

Answer 3

be conscious: the realm of a circle is pi situations the radius squared (not pi squared) Subtract the realm of the circle from the realm of the sq.. Does the circle fill the sq. except for the corners? if so, then the diameter of the circle is two and the radius is 1/2 that or a million. subsequently the realm of the circle is pi situations the sq. of the radius or pi situations 1squared with remains a million (or a million x 3.14159). because you comprehend the realm of the sq. is two x 2 or 4, the realm of the lacking section is 4 - 3.14159 or 0.85841 Does that rationalization make experience to you???