00:01
Hi, in this question we are given with the region, this one, bounded by the two curves, y is equal to 9x square and y is equal to 2, we need to find the centroid of this region.
00:12
First we'll equate the two curves, 2 is equals to 9x square.
00:18
From here we get x square is equals to 2 by 9 or solving this we get x is equal to under root 2 by 3.
00:26
So we get the coordinates for 0 and the closing.
00:30
Using x that is under root 2 upon 3 now first find the area of this region which will be equal to see we'll find this half area and the total will be double so it is equal to two times integration from 0 to under root 2 by 3 and having region the upper curve 2 and the lower 1 is 9 x square so 2 minus 9x square d x now integrating this we get this to be to 2 times 2x minus 9x cube upon 3 and the limits are from 0 to under 2 by 3.
01:10
Now substituting the limit we get this can get be cancelled and this would be equal to 2 times 2 into under 2 by 3 minus 3 times under 2 by 3 raise 2 bar 3.
01:27
Close the bracket solving this further we get this to be equal to two times two root two by three minus two root two by nine solving this further we get two by nine times six root two minus two root two or we can write it as eight root two by nine now we get the area as we know the centroid can be found using the formula which says the coordinates of x is equals to 1 by a integration a to b x times fx minus g x d x and y can be calculated as equal to 1 by a integration from a to b 1 by 2 times fx square minus g x square d x now using these two formulas we can get the coordinates for x and y solving for x first will be equal to 1 by a that is 9 by 8 root 2 integration from 0 to under root 2 and 3 and function x times 2 minus 9x square d x.
03:01
Solving this further, we get equal to 9 by 8 root 2.
03:06
Integration from 0 to root 2 by 3, 2x minus 9x q d x.
03:15
Solving this further, we get this to be equal to 2x square by 2 minus 9 times x -rays to 4 by 4 and the limits are from 0 to root 2 by 3...