Given a curve with parametric equations, x=acos^3(t) and y=asin^3(t), find the length of the curve between points A and B, where t=0 and t=2pi respectively.

The length of an arc between two points on a curve can be calculated in two ways; as the integral of ((dy/dx)^2 + 1)^1/2 between the values of the points, or as the integral of ((dy/dt)^2+(dx/dt)^2)^1/2 between the values of t at the given points. As the curve has been given to us in a parametric form, it would make sense to continue using the second method, as it will save us time.Using the chain rule, we can integrate our function with respect to t. Once we have calculated the derivatives, we want to sum them and take the square root. Noticing that there are common terms in both our expressions for dx/dt and dy/dt, we can simplify the sum, to make it easier to calculate the square root. We now have the expression we would like to integrate, however, it does not look like the nicest function to integrate. Here we have the product of sin(t) and cos(t), which is not immediately a simple integral to solve. We notice that sin(2t)=2sin(t)cos(t), and so we can express this product as a function that only depends on sin, which is much easier to integrate. However, we could also notice that the integral contains both a function and its derivative, (sin(t) and cos(t)) and use a substitution; both will lead us to the same answer. Computing the integral will give us the value of the arc length between the two points on the curve.