Calculate the average over the given interval. f(x)=e^-n x, ...

Key Concepts

Average Value of a Function

The average value of a function over an interval is defined as the integral of the function over the interval divided by the length of the interval. This concept provides a way to summarize the behavior of the function with a single number, representing its typical value across that range.

Definite Integration

Definite integration is the process of finding the integral of a function over a specific interval, which computes the net area under the curve of the function over that interval. It is a fundamental tool for aggregating values, such as in calculating averages or accumulated quantities.

Exponential Functions

Exponential functions are functions of the form e^(kx) where the rate of change of the function is proportional to the function itself. They are important in many areas such as growth and decay processes, and their integration often involves dividing by the constant coefficient in the exponent.

Transcript

00:01 For this question, we're asked to find the average of the function between the interval of negative 1 to 1, where the function is e to the negative nx.

00:12 So to find that average, we're going to use the same equation as we used before, 1 over b minus a times the integral from a to b of f of x, dx.

00:29 So first we're going to plug in all of our values we have.

00:44 So 1 over 1 minus a negative 1 is plus 1 times the integral from negative 1 to 1 of e to the negative n x dx.

01:06 So for this integral, in order to solve it, we're going to have to use u substitution.

01:15 I'm going to just call it u sub.

01:21 And if you don't remember how to do u substitution, how we approach this problem is first we find our u.

01:30 And in this case, our u is going to be negative and x to make this integral a little more easy to solve.

01:38 So we're going to write our u over here is equal to negative and x.

01:47 And then we're going to take the derivative of you.

01:49 So d .u.

01:55 Dx.

01:56 So the derivative of you in respect to x.

01:59 And that's going to equal negative n.

02:07 So now, so we can solve for d .u.

02:10 We're going to multiply each side by dx.

02:21 And these two are going to cancel out.

02:24 And we're going to be left with du equals negative n dx.

02:33 Now the reason why we write this in du dx so that we can solve for du is because in our integral we have dx in there.

02:48 So in order to get rid of the dx and the negative nx, we have to include that in our derivative.

02:57 So this is going to substitute for you.

03:01 So this is going to turn into one half.

03:04 The integral of e to the u, d .u...

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