Question

evaulate an intergral for length of curve with y=(sqrt(4-x^2)) and 0 <_ x <_ 2

im lost, i think i know how to apply the formula but its so difficult to integrate and i need to show all my work!

Answer 1

Looks like a quarter of a circle with radius 2, so the answer is pi.

If you want to show this using integration, you can use:

ds^2 = dx^2 + dy^2 ------>

s = integral of sqrt[(dx/dt)^2 +
(dy/dt)^2] dt

where we have parametrize x and y using parameter t.

In this case it is simplest to choose
x(t) = 2 sin(t) (because we already now the answer: the curve is a quarter circle). The parameter t ranges from zero to pi/2. If you insert this in y=(sqrt(4-x^2), you find y(t) = 2 cos(t) as expected. Then:

sqrt[(dx/dt)^2 + (dy/dt)^2] = 2

If you integrate this from zero to pi/2 you get pi as the answer.