Integral

The integral is the inverse operation of the differential: an infinite sum of infinitesimal pieces. We start from the definite integral as the area under a curve, then we introduce the antiderivative as a tool for computation and the link $\int_a^b f(x)\,dx = F(b)-F(a)$, the heart of the fundamental theorem of calculus. From here we move straight to the geometric applications — volumes of the sphere, cone and pyramid by integration — anticipated by Cavalieri’s principle, the intuitive bridge between synthetic geometry and analysis. Then come the computation techniques (substitution, parts, algebraic fractions, irrational integrals) and the applications of the definite integral: area between two curves, volumes of revolution with the disc and cylindrical shell methods, even and odd functions, improper integrals.

Sezioni

• The definite integral
• The antiderivative
• Table of elementary antiderivatives
• Cavalieri’s principle
• Volumes by integration
• Integration techniques
• Applications of the definite integral

Esercizi

• All the chapter exercises