00:01
So this problem, we have been asked to evaluate a double integral, one with respect to w, one with respect to v.
00:07
Now, when approaching a problem like this, it's always important to know where you're starting.
00:13
Now, if you think back to algebra, when you have nested parentheses, you always go in as far as you can to the innermost set of parentheses, evaluate that and work your way back out again.
00:22
The same holds true for integrals.
00:25
We're going to go to that innermost integral.
00:27
My integral with respect to dw there, we'll evaluate that, then we'll go and evaluate the next one out.
00:35
You always start on the inside first.
00:37
So let's do that.
00:40
What is the value of this innermost integral? well, the variable that we are integrating with respect to is w.
00:50
There is no w shown in this function.
00:53
It's the square root of 1 plus e to the v, not w.
00:57
This whole square root radical, we can treat like a constant.
01:01
So if i was just going to take the integral of 5 with respect to w, it would be 5w.
01:06
If there's an integral of 3 with respect to w, it would be 3w.
01:10
So what i'm going to have here is just the square root times w.
01:16
And i want to evaluate that from w equals 0 to e to the v.
01:24
So let's see what we get when we integrate this.
01:27
If i let w equal e to the v, that gives me e to the v times my radical.
01:33
And if i let w equal zero, it just goes to zero because i'm multiplying by zero in that case.
01:40
So that's what i get...