methods of integration

A great achievement of classical geometry was obtaining formulas for the
areas and volumes of triangles, spheres, and cones. In this chapter we develop a method to
calculate the areas and volumes of very general shapes. This method, called integration, is
a tool for calculating much more than areas and volumes. The integral is of fundamental
importance in statistics, the sciences, and engineering. We use it to calculate quantities
ranging from probabilities and averages to energy consumption and the forces against a
dam s floodgates. We study a variety of these applications in the next chapter, but in this
chapter we focus on the integral concept and its use in computing areas of various regions
with curved boundaries. we saw that a continuous function over a closed interval has a
definite integral, which is the limit of any Riemann sum for the function. We proved that
we could evaluate definite integrals using the Fundamental Theorem of Calculus. We also
found that the area under a curve and the area between two curves could be computed as
definite integrals.
In this chapter we extend the applications of definite integrals to finding volumes,
lengths of plane curves, and areas of surfaces of revolution. We also use integrals to
solve physical problems involving the work done by a force, the fluid force against a
planar wall, and the location of an object s center of mass.