After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and differentiation. Differentiation and integration are readily seen to be reverse operations that compute slopes and under-areas of curves, without requiring tedious infinitesimal limits or infinite summation algebraic procedures. The areas under any two symmetric curves within a square combine to equal its square measure. Corresponding evaluated integrals of any symmetric pair were also found to add to that same area. The general power rules and the fundamental theorems are confirmed for an infinite number of functions containing exponents from the entire real number line, rational or irrational. Any particular equation represents the slope of its own under-area formula, as first discovered by Isaac Newton, where the rate that area accumulates at a point under a curve, traced at constant horizontal velocity, is the value of the curve at that point. Applications of the calculus in mathematics, physics and chemistry elucidated the orbital structure of the atom, vast scientific formula and secrets of the nature of light and gravity.