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.