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.
