I confess to being one of the few who finds those problems interesting. I like the area of a circle proof by limits of triangles, since the calculus proofs assume things about trig functions and about the meaning of pi that are swept under the rug, and at least as difficult as what is being proved.
It also fascinated me to try to generalize and compute the volumes of higher dimensional spheres. when you try it in a way directly analogous to the computation of the usual three dimensional sphere (by slicing) you get an integral with a square root or a n/2 power in it and those are harder than the easy one for the sphere. I.e. the even dimensional spheres are harder by this method than the odd dimensional ones.
Notice in particular that doing a disc (dimension 2) you get a harder integral than for a sphere, but we think we know how to do it because we think we know about trig substitutions and arcsins and so on. Reflect however that even the definition of the sin and arcsin functions actually depends on knowing properties of circles like arc length, which are more difficult than what we are doing. So one should not allow the area of a circle to be computed by integrating (1-x^2)^(1/2) which uses more sophisticated concepts than what they are being applied to.
I ultimately noticed that the easy way to do volumes of 4 spheres and 5 spheres and so on is to use cylindrical shells, since then one gets a simpler integral. This arose when I noticed that the work calculation for pumping water out of a hemispherical pool is actually equivalent to calculating the volume of a 4 sphere by cylindrical shells. (For a disc, this corresponds to sweeping out the disc by a family of expanding circles, or polar coordinates.)
Of course if you want to cheat and throw all the difficulty into the coordinates for the integral, it makes more sense to do the volume of a ball by spherical integration, not polar. I.e. sweeping it out by a family of expanding spheres. But how do you generalize these to dimension 4?