Integrals
Adding up infinitely many thin pieces sounds impossible to finish. How does the integral do it exactly, and why does running a derivative backward turn out to be the key?
Where you are 16 inside
What this area is about
Two questions meet here. Which function has a given derivative? And what exact total do thinner and thinner rectangles approach? The fundamental theorem of calculus says they are one question: for f continuous on [a, b] with antiderivative F, \int_a^b f(x)\,dx = F(b) - F(a).
How these fit together
Antiderivatives. Antiderivatives of powers and of trig, exponential and log functions run derivative rules backward.
The definite integral. Riemann sums estimate with rectangles, the definite integral is their limit, and the fundamental theorem links it to antiderivatives (the last two are areas of their own).
Techniques. u-substitution and integration by parts reverse the chain and product rules. Trig integrals, completing the square, trig substitution, dividing first and partial fractions reshape an integrand into known forms. Choosing a technique picks among them.
Estimates and limits. Trapezoids and Simpson’s rule join the heights at the cut points with segments or parabolas, and usually beat left and right sums. Improper integrals reach to infinity, and their unbounded cousins handle an integrand that blows up.
A path through it
Power antiderivatives, then trig, exponential and log ones. Separately, Riemann sums, then the definite integral; the fundamental theorem needs both strands. Then substitution, parts and trig integrals; completing the square before trig substitution; long division before partial fractions. The trapezoidal rule, then improper integrals over infinite intervals before unbounded integrands. Choosing a technique refers to all of them, so it is last.
Next come applications of integrals and separable differential equations.