Here is a problem that requires nontrivial integration techniques, because the closed form answer is a sum of an algebraic and trigonometric function. If you have ever pulled down blinds by the edge, the parallel slats slant down and make an envelope for a certain curve. In the limit of infinitely dense slats, what is this curve? In case you can't picture this, the curve is defined by the condition that the length of the tangent line from the curve to the y-axis is constant. This is one of the few cases where a reasonably general integral comes out naturally. As for the IVT/MVT, they are not particularly profound. It is in my opinion better to prove things by bisection (which easily proves both, gives intuitive proofs for all their consequences, and essentially is sequential compactness). Bisection was used in 19th century texts, but fell out of favor when the completeness of the reals became standardly axiomatized as the least upper bound principle.