Calculus help?

Question

Find the arc length from (-3,4) clockwise to (4,3) along the circle
x^(2) + y^(2) = 25. Show that the result is one-fourth the circumference of the circle.

How would you find it using the arc lenght method??????

Answer 1

OK, I thought I was familiar with this but it turns out I'm not. Here's how I'd do it.

The length of an arc is given by

L = definite integral from x = -3 to x = 3 of (1 + (f'(x))^2) ^0.5

so in your case you'd solve the above equation for y in terms of x, take the derivative of that function (f'(x)) and plug it in to the integral above and perform the integration. That will give you the length of the arc.

Then calculate the total circumference of the circle using 2*Pi*r

r = sqrt(25)

divide the length of arc you calculated by the circumference and you're done.

Good luck the derivation of the length of arc formula is 3 1/2 pages long in my 1972 text book.

Answer 2

First of all the circle is centered at (0,0) and has a radius of 5. The length of an arc is L=radius*2*angle(radians), or if you prefer to use degrees L=radius*(deg/360)*2*pi. The easiest way to solve the problem is by drawing it out. Looking at the top half of the circle you can create two triangles using the coordinates given, the first has corners at (0,0);(4,3);and (4,0), the second has corners at (0,0);(-3,4);and (-3,0). Find the angles of the triangles (at the origin) using trig and subtract them from 180 to get the angle for your arc. Or, without doing any math at all, if you just look at the triangles, they are both 3x4x5 right triangles, just turned on opposite legs. Therefore, the sum of the two angles must = 90deg (all triangles add up to 180deg) and the angle for the arc is 180-90=90. So using the equations above the total circumference of the circle is 10pi, and the arc length is 5pi*(90/360)=2.5pi

Answer 3

r = 5 from x² + y² = 25
It would be a good idea to make a sketch as follows:-

Let A be point A(4,3) and B be point B(-3,4)

Let AC be perpendicular from A to x axis

sin AOC = 3/5 so angle AOC = 36.9 °

Let D be point (-3,0) and BD is perpendicular to x axis.

tan BOD = 4/3 so angle BOD = 53.1°

Angle BOA = (180 - 36.9 - 53.1)° = 90°

Thus arc AB = 90/360 of circumference = 1/4 of circumference.

Answer 4

Yes.