Question

In Exercises \(11-18\) , find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative. $$y=\int_{-2}^{x} \sqrt{3 t^{4}-1} d t, \quad-2 \leq x \leq-1$$

Short answer

We find the length of the curve by computing the definite integral \( L=\int_{-2}^{-1} \sqrt{1 + (3x^{4} - 1)} dx \). The exact length cannot be calculated without further computations that involve the properties of the square root function and integral computation.

Step-by-step solution

Understand the Problem

We're given the equation \( y=\int_{-2}^{x} \sqrt{3 t^{4}-1} d t \) and the task is to find the length of this curve for the given interval \( -2 \leq x \leq-1 \). To do that, we apply the formula for finding the arc length of a curve \( y=f(x) \) on a specified interval [a, b], which is \( L=\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx \).

Simplify the integrand algebraically before finding an antiderivative

We first find the derivative of the given function. \( f'(x) = \sqrt{3 x^{4}-1} \). Now compute \( (f'(x))^2 \) and simplify. That gives us, \( (f'(x))^2 = 3 x^{4}-1 \). Now we can insert \( (f'(x))^2 \) in the above given formula to further derive the expression.

Inserting the Derived Expression into the Length Formula

We substitute the value we obtained in Step 2 into the formula for arc length. This results in the integral \( L=\int_{-2}^{-1} \sqrt{1 + (3x^{4} - 1)} dx \).

Calculating the Length of the Curve

We now antidifferentiate to get the integral, which gives us the length of the curve. The exact solution depends on the properties of the square root function and basic integral computation rules.