Calculus in the real world?

Question

I have a math project where I have to write a 2 page paper about using calculus applications to solve real-life problems. So far, all the stuff I've found have been too general; I need a specific instance or formula that is commonly used by engineers, physicists, astronomers, economists, etc.. Any ideas? Thanks.

Answer 1

Finding volumes of 3-D solids usually involves calculus.

For a small number of shapes, there are nice formulae. However, most shapes are not so easy.

Consider this example:
For a pyramid with a quadrilateral base, the volume is (1/3)*(Area of base)*height.
If you slice the top off the pyramid, the shape that is left does not have a nice easy formula to calculate its volume. You can get its volume most accurately by setting up a calculus integral.

If you'd like more details about how to do this, feel free to email me. Hope this helps.

Answer 2

Remember, Calculus describes change. Most everything changes. I know some shapes are important for designing video game graphics and things like that, Calculus is used in manufacturing via related rates of something being filled and emptied, models of solids of revolutions are used in tree harvesting to efficiently strip bark from trees in a taper, really there are tons of things out there. I hope this gets your brain thinking.

Answer 3

An example off the top of my head:
Distributed loads in mechanics problems. It's where a force (such as a weight) is distributed over a large area of a beam or some other object. In mechanics, we typically have to draw shear and moment diagrams to find maximum and minimum shear forces or moments, which involve integrating the distributed load to find the shear equation, and then integrating the shear equation to find the moment equation (and for this part, you must also have some known values to find constants). Apart from finding the values of maximums and minimums, this also allows engineers to find the location of the max and min forces or moments acting on the object.

As a materials science major, we also use calculus in thermodynamics. For example, in order to find the change in entropy of something that's being heated, we have to integrate from Temperature 1 to Temperature 2 the specific heat capacity divided by the temperature. (And these problems usually involve having several integrations because specific hat capacity is different over different temperature ranges or the material changes phases). Also in thermodynamics, we use an equation of Euler's (I think it's his) dealing with partial differentiation. I think it's something like
(dz/dx)y = (dy/dx)z + (dz/dy)x (but don't quote me if that's where the x, y, and z's really go... i'm going from memory here) We use this to go from equations that have variables which aren't measurable to equations with all measurable variables.

And in terms of physics, just about anything Maxwell did has a strong base in calculus. (Here's a good start for that http://en.wikipedia.org/wiki/Maxwell's_equations)

Answer 4

differential equations.
area, volume calculations using integral.
perform operations easily by taking logarithm.
finding max, min using derivatives.