# Applications of Integration

Chapter

## 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 where These expressions are equivalent because by the chain rule.

**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$$ z = F\left( y \right)\;and\;y = G\left( x \right)\;so\;z = F\left[ {G\left( x \right)} \right] = H\left( x \right) $$

*H = F*○*G*. Then we have two expressions for*dz*:$$ dz = F'\left( y \right)dy\;and\;dz = H'\left( x \right)dx $$

$$ dy = G'\left( x \right)dx\quad dz = F'\left( y \right)\left[ {G'\left( x \right)dx} \right] = \left[ {F'\left( y \right)G'\left( x \right)} \right]dx = H'\left( x \right)dx $$

## Keywords

Interest Rate Probability Density Function Cylindrical Shell Circular Cylinder Spherical Shell
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1996