Integration and Accumulation of Change
What you must understand
The definite integral is a limit of Riemann sums and represents net accumulation (signed area). Understand left/right/midpoint and trapezoidal approximations and their over/underestimate behavior based on concavity.
The Fundamental Theorem of Calculus has two parts: Part 1, $\frac{d}{dx}\int_a^x f(t),dt=f(x)$; Part 2, $\int_a^b f(x),dx=F(b)-F(a)$ for an antiderivative $F$. Master antiderivatives of power, trig, exponential, and $1/x$ functions, plus properties of integrals (linearity, additivity over intervals, reversing bounds changes sign).
Accumulation functions $\int_a^x f(t),dt$ model net change; the net change theorem connects definite integrals to total change. Use substitution (u-substitution) for composite integrands, including changing limits.
Traps
- Forgetting the chain-rule factor in FTC Part 1 when the upper limit is a function of $x$.
- Ignoring the constant of integration in indefinite integrals.
- Sign errors when reversing limits or integrating below the axis.
- Not adjusting bounds when substituting.
Worked example
$\int_1^4 (2x),dx=[x^2]_1^4=16-1=15$. For $\frac{d}{dx}\int_0^{x^2}\sin t,dt$, FTC Part 1 with chain rule gives $\sin(x^2)\cdot2x$.
Rule of thumb: net area is signed; total distance needs $\int|v(t)|,dt$. Always check bounds after substitution.