This is very standard stuff. Any standard calculus text will demonstrate the techniques and provide very nice graphical depictions. The circumference is more straightforward while the Area can be achieved in (polar, rectangular or parametric variables). Polar coordinatess are the most intuitive since they are 'generalized coordinates' for the given geometry.I'll just articulate the difference
1)The circumference can be computed by rdθ where r is the radius and dθ is an infinitesimal angular displacement of the radius about the center of the circle. This quantity defines the infinitesimal arclength of an infinitesimal sector of the circle. Integration is straightforward by integrating through 2Pi radians to yield (2pi*r) for the circumference or (rθ) for an arclength.
2)The Area requires a double integration. Polar coordinates are still the logical choice although several other techniques yield the same result. The infinitesimal area is now denoted by (dr)rdθ. This captures an infinitesimal radialsector area (infinitesimal in both radius and angle). Integration is straightforward with integration limits of (r=0,R) & (θ=0,2Pi) to yield
A=pi*R^{2}
For the entire area.
A sector may also be calculated by accumulating infinitesimal triangles of area 1/2r(rdθ) and integrating over theta to give the same Area result. However this now also yields the sector area of 1/2r^{2}θ Where θ remains the same [0,2pi]^{
}
Pictures are worth a 1000 words. I'd highly recommend visiting any number of sites to see the infinitesimal arclengths and areas.
3)The solid sphere requires 3 parameters to yield the infinitesimal volume element and uses an additional angle for integration.
1/16/2014

Tom D.
Comments
Both calculate fractions of the whole.
Finding an arc length you find a fraction of the whole circumference of a circle.
Finding the area of a sector you find a fraction of the whole area of a circle.
The fraction in both cases is the item's central angle measure divided by the angle measure of one turn.
E.g., find the arc length and sector area for a central angle of ((3 pi)/2)/(2 pi) = 270°/360° = 3/4 turn of a circle with radius 3 inches.
Arc length = 3/4 * 2*pi*3 = 9 pi/2 inches.
Sector area = 3/4 * pi*3^2 = 27 pi/4 in^2.