00:01
So in this question, we're given an integral between minus 1 and 3 of the square root of 4 minus x minus 1 squared, integrated with respect to x.
00:11
And we are asked to use geometry, as opposed to the riemann sums that you have been using, to evaluate this integral, find the solution.
00:20
And to do that, we need to sketch a graph of the bit inside the integral, which is called the integrand, show which we are integrating.
00:30
And then sort of use that to get our result.
00:34
The first thing to do is to sketch it.
00:37
The easiest way to sketch something is to put numbers into it.
00:43
At first, this function might seem really scary, the square root of 4 minus x minus 1 squared.
00:47
You might be tempted to try and square out that x bracket, but if you just try a couple of points, if you don't notice the trick immediately, it should become quite obvious.
00:55
So if you put in x equals minus 1, then you get minus 1, minus 1, is minus 2.
01:03
Minus 2 squared is 4 and 4 minus 4 is 0 and the square root of 0 is 0.
01:07
So minus 1 is 0.
01:10
If you put in 0, you get the square root of 3, which is sort of there -ish.
01:18
If you put in 1, then you get the square root of 4, which is 2...