Find the arc length of the curve given in parametric form as {x=acos^2t 0<t<2pi y=asin^2t where a is a constant

Answer 1

First, consider the Pythagorean Identity sin^2t+cos^2t=1.

Then we know that:

x+y=(acos^2t)+(asin^2t)=a(sin^2t+cos^2t)=a(1)=a

By subtracting x from both sides, we have:

y=-x+a

which is a line.

Furthermore, sin^2t and cos^2t are non-negative and less than or equal to 1 (from that first identity), so both x and y are between 0 and a.

As a result, the arc length of the curve is the length of the line segment y=-x+a from x=0 to x=a. Together with the x and y-axis, this line segment forms a right triangle (graph it out to see) whose height is |a| and whose length is |a|.

Thus, the length of the curve is the length of the hypotenuse of the right triangle, which we know, by the Pythagorean Theorem, to be the square root of the sum of the squares of the legs, that is, sqrt(a^2+a^2)=|a|(sqrt(2)) (the absolute value is there 'cause length is a positive number and a can be any constant, including a negative number).