00:01
Hi, today we are solving the question in which we are given with the we need to find the centroid of the region bounded by the graph of the function y is equals to x square plus one y is equals to zero x is equals to zero x is equals to three.
00:18
So let us see the graph and the region first.
00:21
So the graph would be like this.
00:23
So as we can see from here, this red line indicates the region x square plus one.
00:29
This blue line indicates x is equals to three.
00:33
This green line indicates y is equals to zero and this purple line indicates x is equals to zero.
00:40
Now from here, finding the centroid of the region, we need to use the formulas that is given by x bar is equals to one by a.
00:55
A is the area limits are from a to b x into f of x minus g of x into dx.
01:05
Similarly the y coordinate is given by one by two into a integral from a to b of f of x square minus g of x square into dx where a is the region fx is the upper function gx is the lower function and a and b are the limits of integration.
01:32
So here in this case upper function f of x is x square plus one g of x is equals to zero limits are a is equals to zero b is equals to three.
01:50
Now substituting the value.
01:53
So finding the area first.
01:55
So area is given by a is equals to integral from zero to three of x square plus one whole into dx.
02:03
Now on solving it, we get it equals to x cube by three plus x and limits are from zero to three.
02:13
Now putting the limits and solving it 27 by three plus three is equals to 12.
02:19
So we get the area.
02:21
Now we will find the x and y coordinates...