Integration and the Fundamental Theorem
Core ideas
Antiderivatives reverse differentiation, and the indefinite integral carries a constant $+C$. The definite integral is a limit of Riemann sums and equals signed area. The Fundamental Theorem of Calculus connects the two: $\int_a^b f(x),dx=F(b)-F(a)$ where $F'=f$, and $\frac{d}{dx}\int_a^x f(t),dt=f(x)$.
Know the power rule for integrals, integrals of $\sin x$, $\cos x$, $e^x$, and $1/x$, and how to use substitution (u-substitution reverses the chain rule). Applications include area between curves, average value, and net change from a rate. Geometry shortcuts help when a region is a known shape.
Traps
- Omitting $+C$ on indefinite integrals.
- Forgetting to change the limits of integration when substituting in a definite integral.
- Misapplying the power rule to $\int x^{-1},dx$, which is $\ln|x|+C$, not a power.
- Computing signed area when total (unsigned) area is requested.
- Dropping the chain-rule factor when integrating composites.
Worked example
Evaluate $\int_0^2 3x^2,dx$. An antiderivative is $x^3$. By the Fundamental Theorem, the value is $x^3\big|_0^2=8-0=8$. Rule of thumb: always find the antiderivative first, then substitute the upper limit minus the lower limit - never subtract before integrating.