00:01
Hello everyone, in this problem, the first part of the problem we need to evaluate the double integral given that double integral over r 7x divided by x square plus y square da given that the region of r as closed interval 1 to 4 cross 0 to 1.
00:21
So now we can rewrite this integral as double integral over r 7x divided by x square plus y square da can be written as integral over x equal to 1 to 4 and integral over y is from 0 to 1 7x divided by x square plus y square dx sorry dy dx.
00:47
So now integrating with respect to y so here it is in the form of 1 plus a square plus y square dy.
00:55
So now taking the 7 out of the integral so it will after integrating that it will be integral over 1 to 4 1 by x multiplied by 1 by x tan inverse of y by x of the limits 0 to 1 dx since integral of 1 by a square plus y square dy to be equal to 1 by a tan inverse of y by a.
01:29
So now substituting the limits upper limit minus lower limit we have this value to be 7 integral of 1 to 4 tan inverse of 1 by x minus tan inverse of infinity multiplied by dx.
01:52
So now this tan inverse of infinity will become pi by 2 so now substituting that value so 7 into integral of 1 to 4 tan inverse of 1 by x minus pi by 2 dx.
02:10
Now integrating with respect to x so we will be having that to be 7 of now we can separate the integral so it will be integral 1 to 4 tan inverse of 1 by x minus sorry dx minus 7 into integral of 1 to 4 pi by 2 dx.
02:35
So integrating this we will be having this to be value of 7 of log of x square plus 1 divided by 2 plus x tan inverse of 1 by x of the limits 1 to 4 plus 7 pi by 2 of x of the limits 1 to 4.
03:01
Now substituting the limits upper limit minus lower limit we have the value to be 7 multiplied by log of 17 by 2 plus 4 tan inverse of 1 by 4 minus log of 2 by 2 minus pi by 4 plus 7 pi by 2 multiplied by 3 sorry i forgot to mention the bracket.
03:37
So now simplifying this we have the value to be 7 by 2 log of 17 by 2 plus 28 tan inverse of 1 by 4 plus 35 pi by so therefore this is the value of the given integral double integral of over r 7x divided by x square plus y square dx.
04:13
So this is the required answer...