Applications of integrals

What do area, length, volume, work, and hydrostatic force have in common? All of these (and many other important concepts in science and engineering) can be modelled as Riemann sums (8.6) $$\sum\limits_{k = 1}^n {f\left( {{\xi _k}} \right)} {\rm{ }}\Delta {x_k},$$ and computed as integrals (8.28), $$\int\limits_{a}^{b} {f(x)dx: = \mathop{{\lim }}\limits_{{\parallel \mathcal{P}\parallel \to 0}} } \sum\limits_{{k = 1}}^{n} {f({{\xi }_{k}})\Delta {{x}_{k}}.}$$ In this chapter integrals are applied to problems of computing areas (Sects. 11.1, 11.2, and 11.6), arc lengths (Sect. 11.3), volumes (Sects. 11.4 and 11.5), moments and centroids (Sects. 11.7 and 11.8), work (Sect. 11.9), and hydrostatic force (Sect. 11.10).