Find the length of the curve given by the equation y= intergral from -pi to x of sqrt(cos(t)) dt for x between -pi and pi. I think I know to do this- at least part of it. I am using the fundamental theorem of calculus and the arc length formula, but I don't know how to find the anti derivative of sqrt(cos(t)).

Camryn/Kira/Dean -- please use the same name for your posts.

That antiderivative is not expressible in elementary functions. It is an elliptic integral.

Ok but then how would I solve this problem?

you'd have to use some numeric integration technique to get an approximation.

To find the length of the curve given by the equation y = ∫√(cos(t)) dt from -π to x, you are correct in using the fundamental theorem of calculus and the arc length formula.

The first step is to find the antiderivative or indefinite integral of √(cos(t)) with respect to t. Unfortunately, there is no elementary antiderivative for √(cos(t)) in terms of elementary functions.

However, there is a special function called the elliptic integral that can represent the antiderivative of √(cos(t)). In particular, the antiderivative of √(cos(t)) is expressed using the elliptic integral of the second kind.

The elliptic integral of the second kind, denoted as E(·, ·), is defined as:

E(φ, k) = ∫√(1 - k²sin²(φ)) dφ,

where φ is the angle and k is the modulus. In our case, we have √(cos(t)) as our integrand, so we need to find a way to convert it to the form of the elliptic integral.

To do this, we use the substitution cos(φ) = cos(t). From there, we can express the integrand as √(cos(t)) = √(cos²(φ)) = cos(φ).

Substituting these values, our integral becomes:

∫ cos(φ) dφ.

Now we can see that this is the same form as the elliptic integral, with φ as the angle and k = sin(π/2) = 1 as the modulus.

Hence, our integral becomes:

E(φ, 1) = ∫√(cos(t)) dt.

However, to evaluate this integral, we require numerical methods or special functions. You can use software programs or calculators that have built-in functions for elliptic integrals to obtain the antiderivative of √(cos(t)).

Once you have the antiderivative, you can use the arc length formula to find the length of the curve:

L = ∫√(1 + (dy/dx)²) dx,

where dy/dx is the derivative of y with respect to x.

In this case, y is given as the antiderivative of √(cos(t)). Therefore, you need to differentiate the antiderivative to find dy/dx and then substitute it into the arc length formula.

You can approximate the length numerically using numerical integration methods or seek software assistance to evaluate this integral and find the length of the curve.

Note: The elliptic integral and its derivatives involve complex mathematical concepts beyond the scope of this explanation.