00:01
Okay, so first of all, let's sketch the region enclosed by our two curves.
00:08
So this is the x -axis, this is the y -axis.
00:12
Now we have x plus y equals 56, so 56 here, 56 here.
00:20
And this is our line, x plus y, equals 56.
00:29
Then we have y equals x squared which is just a parabola.
00:37
Oh maybe looking, okay, perfect.
00:40
Now we have our parabola, which looks like this one here.
00:47
So this is y equals x squared.
00:50
Now the first thing that we need to do is this.
00:53
We need to find these two intersection points.
00:56
So here we need to solve the question.
01:01
56 minus x which is what we get from x plus y equals 56 equals x squared now this equation is equivalent to x squared plus x minus 56 equals 0 okay perfect let's solve this equation by using the quadratic formula we have x equals negative 1 plus or minus square root of what we are going to have 1 plus 4 multiplied by 56 over 2 okay so this one is negative 1 plus or minus square root of what okay we are going to have square root of 25 which is 15 so here what do we get? we get two solutions, negative 8 and 7.
02:05
Okay, perfect.
02:06
So the x coordinate of this point is negative 8.
02:10
The x coordinate of this point is 7.
02:13
Okay, now we need to find the area of this region.
02:19
Well, what is the area of this region? this is going to be an integral from negative 8 to 7 of, what okay and 56 minus x minus x squared in the x perfect and well this guy is what well this guy is an integral actually we can already compute an antiderivative so this one is going to be 56 x minus x squared over 2 minus x cubed over 3 evaluated between negative 8 and 7 so what are we going to have 56 multiplied by 7 minus minus 8 so 15 minus 1 half multiplied by 7 squared minus 8 so 15 minus 1 half okay let me write like this 49 minus 64 minus one third multiplied by seven cubed seven cubed plus eight cubed okay perfect let's see if we did everything correctly okay yes this is correct okay perfect now let's see go on well we need to find the x coordinate of the centroid and what is the x coordinate of the centroid well this one is gonna be one over the area of our region that we computed a so 1 over a multiplied by okay so here we are gonna have let's see a double integral over our region so let me call are this region.
04:28
Okay, so i was saying a double integral over our region are of x, the a, the infinitesimal element of area...