00:01
Hello students, for the path a to find the arc length from minus 3, 4, 2, clockwise to 4, 3 along the circle defined x square plus y square equal to 25 minus 3, 4, 2, 4, 3.
00:15
We can use the arc length formula that is arc length equal to integration from a to b in square root 1 plus dy by dx whole square into dx.
00:30
First let's equation of the circle that is so first let's find the equation of the circle that is x square plus y square equal to 25.
00:42
Now, let's derivate dy by dx 2x plus 2y dy by dx equal to 0.
00:49
Simplifying will get dy by dx equal to minus x by y.
00:54
Now we can substitute dy by dx into the arc length formula.
00:57
So arc length equal to integration from a to b square root 1 plus minus x by y whole square into dx to find the limits of integration a b.
01:13
We need to determine the starting and ending angles along the circle.
01:16
The point minus 3 by 4 lies in second quadrant and the point 4, 3 lies in first quadrant.
01:34
The angle between the positive x -axis and the line joining the center of the circle to the point can be found using the arc tan that is theta 1 equal to arc tan minus 4, 4 by minus 3 that is minus 53 .13 degree.
02:00
Similarly, the angle between the positive x -axis and the line joining the center of the circle of the point 4, 3 can be found that is theta 2 that is arc tan 3 by 4 36 .87 degree.
02:16
Since we are moving clockwise theta 2 is the starting angle, theta 2 is starting angle and theta 1 will be the ending angle.
02:35
So theta 2 will be a and theta 1 will be b.
02:40
Now we can substitute the value into the arc length formula and evaluate the integral...