Find the length of the arc of 𝑥^2 + 𝑦^2 = 64 from 𝑥 = −3
to 𝑥 = −1, in the second quadrant.

We are tasked with finding the length of the arc of the circle given by the equation \( x^2 + y^2 = 64 \) from \( x = -3 \) to \( x = -1 \) in the second quadrant.

### Step 1: Recognizing the Equation
The equation \( x^2 + y^2 = 64 \) represents a circle with radius \( r = 8 \) centered at the origin.

### Step 2: Parametric Form of the Circle
We can parametrize the circle using:
\[
x = 8 \cos(\theta), \quad y = 8 \sin(\theta)
\]
where \( \theta \) is the angle from the positive \( x \)-axis.

### Step 3: Find the Angle Corresponding to the Given \( x \)-Values
For \( x = -3 \):
\[
x = 8 \cos(\theta) \quad \Rightarrow \quad -3 = 8 \cos(\theta) \quad \Rightarrow \quad \cos(\theta) = -\frac{3}{8}
\]
This gives:
\[
\theta_1 = \cos^{-1}\left(-\frac{3}{8}\right)
\]

For \( x = -1 \):
\[
x = 8 \cos(\theta) \quad \Rightarrow \quad -1 = 8 \cos(\theta) \quad \Rightarrow \quad \cos(\theta) = -\frac{1}{8}
\]
This gives:
\[
\theta_2 = \cos^{-1}\left(-\frac{1}{8}\right)
\]

### Step 4: Arc Length Formula
The length of the arc of a circle is given by:
\[
L = r \Delta \theta
\]
where \( r \) is the radius and \( \Delta \theta = \theta_2 - \theta_1 \) is the difference in angles. Here, \( r = 8 \), so:
\[
L = 8 (\theta_2 - \theta_1)
\]

### Step 5: Calculate the Arc Length
Now we need to find the numerical values of \( \theta_1 \) and \( \theta_2 \), and then compute the arc length.

Let me calculate this.It seems the calculated arc length came out negative, which doesn't make sense for a physical length. This happened because the angles are measured from the positive \(x\)-axis and we're dealing with the second quadrant. We need to adjust for the fact that the circle is traversed in the counterclockwise direction in the second quadrant.

Let me correct this and recalculate the arc length properly.The length of the arc from \( x = -3 \) to \( x = -1 \) in the second quadrant is approximately 2.07 units.

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions to consider:
1. How would the arc length change if the radius of the circle were different?
2. Can you calculate the total circumference of the circle?
3. What is the equation for the arc length of a curve defined by a more general function \(y = f(x)\)?
4. How would you find the arc length if the circle were centered at a different point?
5. How do parametric equations help in determining geometric properties of shapes?

**Tip**: When dealing with circles, always check whether you're working in radians or degrees, especially when using trigonometric functions.

Find the length of the arc of 𝑥^2 + 𝑦^2 = 64 from 𝑥 = −3 to 𝑥 = −1, in the second quadrant.

Explore how to calculate the arc length of a circle with equation x^2 + y^2 = 64 from x = -3 to x = -1 in the second quadrant. This problem covers key geometric and trigonometric concepts, including parametric equations, the arc length formula, and the use of inverse trigonometric functions. Suitable for high school students in Grades 10-12.

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