00:01
For this problem, we are asked to use an iterated integral to find the area of the region bounded by the graphs of the equations, y equals 4 minus x squared and y equals x plus 2.
00:09
So the first step that we want to take here is figure out what the points of intersection of the two lines would be.
00:15
Specifically, we want to have x plus 2 equal 4 minus x squared.
00:20
So that would be the case when we have that x plus x squared equals 4 minus 2.
00:24
So x plus x squared equals 2.
00:27
So we can actually solve this as a quadratic, or let me see here, we can solve this, write this as x squared plus x minus 2 equals 0.
00:42
So we'd have that that would be the case, if we factor this, we can write this as x plus 2 times x minus 1 equals 0.
00:52
So we have solutions when we have x equals negative 2 and x equals positive 1.
00:57
Then, now i've plotted this out as well just to see how things shake out here.
01:04
So we can see that we'd have the lower boundary for our region that we're trying to integrate is that straight line x plus 2, and the upper boundary is that 4 minus x squared.
01:14
So we can write our area or our iterated integral for the area as being the integral from negative 2 up to 1 of the integral from x plus 2 up to 4 minus x squared, d .y, dx, which will then be equal to the integral from negative 2 up to 1 of 4 minus x squared minus bracket x plus 2, so that becomes minus x minus 2, the x.
01:42
Or we can write this as the integral from negative 2, oops, not negative 1, negative 2, up to one of four, or not four anymore, because we have that minus two...