13.4 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

## 13.4 Applications of calculus (optional calculus-based section)

The student who has studied integral calculus will recognize that the graphical rule given in the previous section can be reexpressed as an integral,

`W=int_(x_1)^(x_2) Fdx.`

We can then immediately find by the fundamental theorem of calculus that force is the derivative of work with respect to position,

`F=(dW)/(dx).`

For example, a crane raising a one-ton block on the moon would be transferring potential energy into the block at only one sixth the rate that would be required on Earth, and this corresponds to one sixth the force.

Although the work done by the spring could be calculated without calculus using the area of a triangle, there are many cases where the methods of calculus are needed in order to find an answer in closed form. The most important example is the work done by gravity when the change in height is not small enough to assume a constant force. Newton's law of gravity is

`F=(GMm)/r^2`,

which can be integrated to give

`W=int_(r_1)^(r_2) (GMm)/r^2dr`

`=-GMm(1/r_2-1/r_1)`.

13.4 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.