Remember that the area under a curve has to be bounded, so limits need to be applied to define the boundaries. Usually it's the area between the curve and one axis or both. Occasionally it's the area between two curves or the area produced by the intersection of two curves. Sometimes, as in the case of a circle or ellipse which already encloses an area, it's the whole or part of an area inside the curve. The "formula" is based on the area of very thin rectangles, infinitely thin, in fact, which are laid side by side to fill the area. An integral is applied which sums the areas of the rectangles over the whole region specified to get the area of that region. The most common formula is integral(ydx) where y=f(x) defines the curve. Limits of x are then given to enclose the area, so that definite integration is applied. The best way to approach any problem involving finding areas related to a curve is to sketch the curve and draw the area that needs to be found. Imagine a large curve had been drawn on the ground and you had a roll of sticky tape. You work out where the area is and cut strips of tape and stick them down so that you fill the required area. You can only lay the strips side by side, no criss-crossing and no laying the tape in a different direction. You can only cut the strips into rectangles using a cut straight across, not at an angle. You end up with not quite filling, or slightly over-filling the space because the tape will overlap the curve slightly. But when you step back it will look like the area has been properly covered by tape. If you used narrower tape the area would be even better filled. That's the principle on which the integration is based, because the area of each rectangle is the length of the strip y times the width we call dx (the width of the strips of tape).
The area outside a circle must be bounded. You need the equation of the circle so that you can relate x and y. For example, the general equation of a circle is (x-h)^2/r^2 + (y-k)^2/r^2=1, where r is the radius and (h,k) is the centre. So y=k+sqrt(r^2-(x-h)^2) is half the circle, because the other half is k-sqrt(r^2-(x-h)^2). If you need the area between the circle and the x axis between a and b, you need the lower part of the curve given by the second expression; if you need the upper part of the curve you need the first expression. If the circle is coloured red and the outside of the circle is blue, the area between the lower circle and the x axis will be entirely blue, whereas the area between the upper circle and the x axis will be red and blue. Get the picture? Once you've decided what you want, you compose the integral: integral(ydx)=integral((k-sqrt(r^2-(x-h)^2))dx) for b<x<a, for example. After integration you apply the limits by substituting x=b into the expression and subtracting the expression when x=a.