CHAPTER 05

Integration

Understand integration as accumulation and as the inverse relationship to differentiation.

\[ \text{small contributions} \rightarrow \text{accumulation} \rightarrow \int \rightarrow \text{total change} \]

INTEGRATION

From accumulation to applications

Understand integration as the accumulation of many small contributions.

Reverse differentiation and understand indefinite integrals and the constant of integration.

Study accumulated change over an interval, signed area and evaluation at limits.

Connect differentiation, antiderivatives and accumulated change.

Reverse the chain rule by introducing a suitable change of variable.

Reverse the product rule to integrate products of functions.

Rewrite rational expressions into simpler fractions that can be integrated.

Use limits when an interval is infinite or the integrand becomes unbounded.

Use integration for areas, accumulated rates, displacement, average values, volumes and probability.

\[ F'(x)=f(x) \qquad\Longleftrightarrow\qquad \int_a^b f(x)\,dx=F(b)-F(a) \]

Differentiation describes local change. Integration accumulates local change. The Fundamental Theorem of Calculus connects these two ideas.