Calculus pp 181-207

# Applications of Integration

• Harley Flanders
Part of the Textbooks in Mathematical Sciences book series (TIMS)

## Abstract

The purpose of Integration is to “add up” small quantities. In this chapter we apply Integration to a variety of situations. First we introduce differentials, which are intuitive and a big help in setting up problems. They are the quantities that appear under the integral sign, like F(x)dx. If we have two variables x and y related by a function y = F(x), then we write dy = F’(x)dx. Because of the chain rule, differentials have an inner consistency. For instance, suppose
$$z = F\left( y \right)\;and\;y = G\left( x \right)\;so\;z = F\left[ {G\left( x \right)} \right] = H\left( x \right)$$
where H = FG. Then we have two expressions for dz:
$$dz = F'\left( y \right)dy\;and\;dz = H'\left( x \right)dx$$
These expressions are equivalent because
$$dy = G'\left( x \right)dx\quad dz = F'\left( y \right)\left[ {G'\left( x \right)dx} \right] = \left[ {F'\left( y \right)G'\left( x \right)} \right]dx = H'\left( x \right)dx$$
by the chain rule.

## Keywords

Interest Rate Probability Density Function Cylindrical Shell Circular Cylinder Spherical Shell
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