Multiple Integrals

Multiple Integrals

      Multiple Integrals are basically a continuation of regular integrals that we already went over in AP Calc. Throughout the year, we reviewed only integrals that have a single variable. Multiple integrals have more than one variable, usually 2 or 3. 

     Instead of calculating the area under the curve of a function on a 2-dimentional plane, an integral with 2 variables calculates the volume that a certain function makes on a 3 dimensional plane, as shown below:

Since an integral with 2 variables can be used to calculate the 3rd dimension, Triple integrals can be used to calculate a 4th dimension, such as time or density.

We can talk all we want about what integrals are used for, but it is easier to understand just by doing a couple examples:

We will start with a simple example,

When 

    We can check our work by flipping the limits on the integral, and then also flipping the order of integration. But you always have to Integrate inside out. 

Bingo! The answers are the same.

   Next, we will try integrating this same function, except our limits will be variables instead of constants. This seems complicated, but is actually quite easy.

   To tackle multiple integrals, remember to take it one step at a time until you feel comfortable speeding up.

To watch another example by Patrick JMT, watch

http://www.youtube.com/watch?v=jSjFKcOZY3s

Enjoy your summer(s) Calculus studs

Integration by Trig Substitution

Blog Post on google docs because I am not a fan of the Blogger's formatting. 

https://docs.google.com/document/d/1E5TqLG3DPgaYTAACS_1_mfq2iioFC9CYd3c6V6NFvpc/edit?usp=sharing

Arc Length of a Curve!

The arc length of a curve is a fairly simple concept. We've already learned how to find the arc length of circles, but what about functions like cos(x+5) . . . Well first let's define arc length:

The length of a curve if were to be “rectified”.

"Rectified" is simply straightening out a curve from it's "curvy" shape to a linear line that can be measured. Now, you could do this with string, but you can also do it with calculus. Before we get into the calculus, here is a link to visualize a rectified curve: click here

So here is the calculus–again, nothing you don't already know how to do. Since, we're only going to go over how to find the arc length rectangularly

The first step is having a function that you would like to have the , and the formula.

For specifically rectangular form, ds is replaced with a formula showed below:

Since you already know how to find derivatives, the process is fairly simple from here on out. You can either use your graphing calculator by using "fnINT" of the rectangular formula above, or you can use Geogebra! Below is a 10-minute video I made to show how to take a real life object, and find the arc length of it. Not surprisingly, solving with calculus is much easier than measuring a curve with string.

If you would like to check out some examples of how to solve polar functions or parabolic functions, check out this website: click here. The formulas are also fairly simple, and easy to solve based on the calculus you know. Adios amigos.

Eliot

Mathematics of the Natural World

Patterns in nature are visual regularities of form that occur in the natural world. These patterns can be modeled with mathematics and physics. Natural patterns can include symmetries, fractals, spirals, meanders, waves and dunes, foams and bubbles, arrays, cracks, and stripes (some examples are shown below):

Romanesco Broccoli in fractal form

radial symmetry

bilateral symmetry in a zebra's stripes

radial symmetry in a kiwi

logarithmic spiral of a Nautilus

dune meanders

tessellation array of scales

tessellation array of scales

sand dunes at equivalent angles

crack patterns

inelastic crack patterns

meanders

phyllotaxis Fibonacci spirals

phyllotaxis arrangement

Philosophers, mathematicians, and physicists have applied their skills to the natural world across the ages. Early Greek philosophers Plato, Pythagoras, and Empedocles often studied natural form, hoping to explain the ordered occurrence of patterns. In the 19th century, Joseph Plateau developed the theory of minimal surface area as shown in soap bubble films, and was able to mathematically model the concept. Ernst Haeckel painted thousands of Radiolaria (small marine organisms) to show their symmetry in detail. D'Arcy Thompson extensively studied and modeled the growth patterns of flora and fauna, applying mathematics to spiral growth. Alan Turing established methods for predicting morphogenesis in embryos that would eventually become spots and stripes. Benoit Mandelbrot and Aristid Lindenmayer developed the concept of fractals that can be used to approximate plant growth patterns.

While the models are not always spot on, the conceptual process of predicting the patterns of the natural world has broadened our horizons and increased our appreciation of the beauty of nature.

Partial Differentiation

Partial Differentiation

Earlier in the year we learned how to differentiate a function.Today we took that a step further by talking about PARTIAL DIFFERENTIATION!!! Basically, partial differentiation is the process of finding the derivative in regards to a single term–either x or y. This process can be used when a function has more than one variable. Before we can take the partial derivative of a function, we need to learn about some new notation. denotes that a function is in terms of both the x and y denotes that a partial derivative needs to be taken in terms of x denotes that a partial derivative needs to be taken in terms of y

Likewise, the partial derivative notations above can be written as  and  or as  and .

For the majority of first and second semester we used limit definitions to understand how and why derivatives work. Even though limits are not the most efficient way to find derivatives, they’re important when understanding them fully. Remember that derivatives have a limit definition of . The definition of a partial derivative using limits can be split into both an x and a y component, but both follow the basic structure of the original limit definition. However, each variable is approaching a separate number. As h approaches zero, x approaches a and y approaches b. This is written as:

 and

Notice, that in the numerator, h is placed with a, since that is the value x is approaching. Likewise, when taking the partial derivative of y, h is placed with b, since that is the value that y is approaching. Basically what the limit definition explains is that partial differentiation requires that you only take the derivative with regards to one variable at a time and leave the other as a constant. That really is the most important thing to keep in mind when finding partial derivatives. ALWAYS treat the variable that you’re NOT taking the partial derivative of as a constant. For example...

Before getting into higher order partial derivatives, it’s also important to understand the geometric representation of a partial derivatives. In this case we’re looking at a three-dimensional surface, in the x, y, z axes. Just as the derivative of a function represents the slope of a tangent line to  at ,  and  are the slopes of tangents lines. However, there’s a difference. They are slopes of traces of surfaces, which are curves that represent the intersection of the surface and the plane given by  or . That is, when taking the derivative in terms of y, it is the slope of the tangent plane in the y direction. Likewise, the derivative in terms of x represents the slope of the tangent plane in the x direction. Later, as we talk about higher order partial derivatives, you’ll see that when you take a derivative in terms of both x and y, it’s geometric representation can be described as the rate of change of the slope in the x-direction as one moves in the y-direction. In order to understand these concepts more clearly, this applet will be helpful.

Taking higher order partial derivatives is relatively easy. You simply take the partial derivative in terms of either x or y, and then take the partial derivative in terms of x or y again, depending on the problem. Here’s an example:

Just like there were differentials for functions with one variable, there are also differentials for functions with multiple variables. We learned this year that differentials were:

With multiple variables, this becomes written as:

Those two differential equations above are synonymous. What this is saying is that the differential of a function with two variables is just equal to the sum of the partial derivative of x and y multiplied by a change in x and change in y respectively. This can likewise be expanded if you had a function with more than two variables, such as :

If you’re still feeling a bit stuck, here are two PatrickJMT videos to refer to!

Partial Differentiation and Higher Order Partial Differentiation