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Calculus: Integration

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Calculus: Integration

This module reviews the basic principles of mathematics covered in the FE Exam. We first review the equations and characteristics of straight lines, then classify polynomial equations, define quadric surfaces and conics, and trigonometric identities and areas. In algebra we define complex numbers and logarithms, and show how to manipulate matrices and determinants. Basic properties of vectors with their manipulations and identities are presented. The discussion of series includes arithmetic and geometric progressions and Taylor and Maclaurin series. Calculus begins with definitions of derivatives and gives some standard forms and computation of critical points of curves, then presents grad, del and curl operators on scalar and vector functions. Differential equations are calcified and to methods to solve linear, homogenous equations are presented. Fourier series and transforms are defined along with standard forms, and finally Laplace transforms and their inverse are discussed. In all cases, basic ideas and equations are presented along with sample problems that illustrate the major ideas and provide practice on expected exam questions.Time: Approximately 4.5 hours | Difficulty Level: Medium

The purpose of this course is to review the material covered in the Fundamentals of Engineering (FE) exam to enable the student to pass it. It will be presented in modules corresponding to the FE topics, particularly those in Civil and Mechanical Engineering. Each module will review main concepts, illustrate them with examples, and provide extensive practice problems.

Taught By

Dr. Philip Roberts
Professor

Transcript

Continuing our discussion of calculus, now I'd like to talk about integration,

and integration, of course, is the inverse or the reverse of differentiation.

And it's defined here in this extract from the handbook as, in finite form, the limit

as n To infinity of the summation of the function of x multiplied by delta x.

Or, in the limit, as n goes to zero,

the summation sign turns into an integral sign.

Where this becomes the integral of the limits of a to b of function of x, dx.

Now again normally we won't have to calculate these first principles and

here is the extract from the table again That I showed previously, and

the column on the right-hand side here is integrals, shows integrals

of certain common functions, which we can refer to for particular questions.

And furthermore,

we talk about a definite integral as being an integral over a finite range.

For example in this case an integral over the limits from a to b.

In which case the integral becomes the area under the curve

of f of x between those limits.

Or the integral is indefinite in other words the limits are unrestricted.

For example The area under the curve of the function y is equal

to cosine pi x over the range x from 0 to 0.5 is which of these alternatives?

So, here is what we're doing.

Here is the graph of this function.

Y is equal to cosine pi x, and

we are asked to evaluate the area of the curve from 0 to 5.

In other words the area of this portion of the curve, right here.

So we use our basic formula that the area is the integral of

the limits of f of x dx and in this case then,

the limits of the integration Our ranges from 0 to 0.5.

Function of x is cosine pi x.

And we can look up this integral or evaluate it fairly easily from the table,

and the integral of cosine pi x is 1 over pi, sine pi x, and

in this case, we're evaluating this function at 0 over the range of 0 to 0.5.

So that is, therefore, equal to 1 over pi sine pi

over 2 times 0.5 minus sine 0 at the lower limit.

And sine pi by 2 is 1.

Sine 0 is 0.

So therefore, the answer is 1/pi and the answer is b.

And just as a note, if this was expressed as an indefinite integral,

for the example integral cos pi xdx, The answer would be

one over pi sin pi x but then we have to add some unknown constant

of integration to that if the integral is indefinite.

Another useful rule is L'Hospital's rule Which

allows us to evaluate the limit of a value,

of a quotient of two numbers, which is undefined.

So, for example, if I want to determine the limit of

a function, f of x divided by g of x.

As X approaches A, where both of those functions approach zero or infinity.

In other words, zero divided by zero, or

infinity divided by infinity is impossible, it's undefined.

So,

And let me illustrate that by means of an example.

Suppose I want to get the limit of this function sin x/x as x tends to 0.

Well, that’s undefined because sin of 0, and x of 0 is also 0.

So, that would be 0 divided by 0, which is undefined.

If I apply L'Hospital's rule to that, however, and

differentiate the top and the bottom, the differential of sine x is cosine x.

Differential of x is 1.

Therefore, this tends in the limit to cosine x, and

if I now substitute x equals to 0 in that We see that the limit of

the function sine x over x as x goes to zero is one.

Now, I'll also mention briefly in this section,

they include centroids and moments of inertia which are basically

applications of integration to various areas However,

I'm not going to cover it here because we'll do this in more detail

later in module six, Statics, and module eight, Mechanic of Materials.

So, this concludes some elementary properties of integration.

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