To find the length of the line given by the equation $y = -16x + 6$ from the point $(-1, 22)$ to the point $(1, -10)$, we will use the arc length formula for a curve $y = f(x)$:

$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

Step 1: Find the derivative $\frac{dy}{dx}$.

The function is $y = -16x + 6$. The derivative with respect to $x$ is:

$\frac{dy}{dx} = -16$

Step 2: Substitute the derivative into the arc length formula.

We substitute $\frac{dy}{dx} = -16$ into the formula:

$L = \int_{-1}^{1} \sqrt{1 + (-16)^2} \, dx$

Step 3: Simplify the integrand.

Simplifying inside the square root:

$L = \int_{-1}^{1} \sqrt{1 + 256} \, dx = \int_{-1}^{1} \sqrt{257} \, dx$

Step 4: Integrate.

Since $\sqrt{257}$ is a constant, the integral simplifies to:

$L = \sqrt{257} \int_{-1}^{1} dx = \sqrt{257} \cdot [x]_{-1}^{1}$

Evaluating the integral:

$L = \sqrt{257} \cdot (1 - (-1)) = \sqrt{257} \cdot 2$

Final Answer:

The length of the line is:

$L = 2\sqrt{257}$

Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:

1. How does the arc length formula change if the curve is given in parametric form?
2. What happens to the arc length if the function is non-linear?
3. Can we compute the arc length of a curve in 3D space?
4. What is the geometric interpretation of the arc length for a line?
5. How do we find the arc length if the function is not differentiable?

Tip: When calculating the arc length, always make sure the function is continuous and differentiable in the interval of interest. This ensures that the derivative exists and the arc length can be accurately computed.