Refer to the graph and information: An ant is crawling on a straight wire. The velocity, v(t), of the ant at time 0 ≤ t ≤ 8 is given in the graph. Note: The graph on 0 ≤ t ≤ 2 is a semicircle (half-circle).The curve end points are at (0, 0) and (2, 0) with vertex at (1, 1). Line 1 begins at (2, 0) and rises right till (4, 3). Line 2 begins at (4, 3) and falls right till (6, 0) in quadrant 1. Line 3 begins at (6, 0) and fall right till (7, minus 1) in quadrant 4. Line 4 begins at (7, minus 1) and rises right till (8, 0). What is the total distance traveled by the ant over the time interval 0 ≤ t ≤ 8?

To find the total distance traveled by the ant over the time interval \( 0 \leq t \leq 8 \), we need to compute the integral of the absolute value of the velocity function \( v(t) \) over this interval. The total distance is given by:

\[
\text{Total Distance} = \int_{0}^{8} |v(t)| \, dt
\]

### Description of the Velocity Function

1. **Semicircle from \( t = 0 \) to \( t = 2 \)**:
   - The velocity function \( v(t) \) forms a semicircle from (0, 0) to (2, 0) with a maximum point at (1, 1).
   - The equation of the semicircle can be described as \( v(t) = \sqrt{1 - (t - 1)^2} \) for \( 0 \leq t \leq 2 \).

2. **Line Segment from \( t = 2 \) to \( t = 4 \)**:
   - The velocity increases linearly from 0 to 3.
   - The equation of this line segment is \( v(t) = \frac{3}{2}(t - 2) \).

3. **Line Segment from \( t = 4 \) to \( t = 6 \)**:
   - The velocity decreases linearly from 3 to 0.
   - The equation of this line segment is \( v(t) = 3 - \frac{3}{2}(t - 4) \).

4. **Line Segment from \( t = 6 \) to \( t = 7 \)**:
   - The velocity decreases linearly from 0 to -1.
   - The equation of this line segment is \( v(t) = -t + 6 \).

5. **Line Segment from \( t = 7 \) to \( t = 8 \)**:
   - The velocity increases linearly from -1 to 0.
   - The equation of this line segment is \( v(t) = t - 8 \).

### Calculation of the Distance

We will integrate the absolute value of the velocity for each segment.

1. **Semicircle (0 to 2):**
   \[
   \int_{0}^{2} \sqrt{1 - (t - 1)^2} \, dt
   \]

2. **Line Segment (2 to 4):**
   \[
   \int_{2}^{4} \left| \frac{3}{2}(t - 2) \right| \, dt = \int_{2}^{4} \frac{3}{2}(t - 2) \, dt
   \]

3. **Line Segment (4 to 6):**
   \[
   \int_{4}^{6} \left| 3 - \frac{3}{2}(t - 4) \right| \, dt = \int_{4}^{6} \left( 3 - \frac{3}{2}(t - 4) \right) \, dt
   \]

4. **Line Segment (6 to 7):**
   \[
   \int_{6}^{7} \left| -t + 6 \right| \, dt = \int_{6}^{7} (-t + 6) \, dt
   \]

5. **Line Segment (7 to 8):**
   \[
   \int_{7}^{8} \left| t - 8 \right| \, dt = \int_{7}^{8} (8 - t) \, dt
   \]

Now, let's compute each integral:

#### 1. Semicircle (0 to 2)

\[
\int_{0}^{2} \sqrt{1 - (t - 1)^2} \, dt = \frac{\pi}{2}
\]

#### 2. Line Segment (2 to 4)

\[
\int_{2}^{4} \frac{3}{2}(t - 2) \, dt = \left[ \frac{3}{4}(t - 2)^2 \right]_{2}^{4} = \frac{3}{4}(2)^2 = 3
\]

#### 3. Line Segment (4 to 6)

\[
\int_{4}^{6} \left( 3 - \frac{3}{2}(t - 4) \right) \, dt = \left[ 3t - \frac{3}{4}(t - 4)^2 \right]_{4}^{6} = 6
\]

#### 4. Line Segment (6 to 7)

\[
\int_{6}^{7} (-t + 6) \, dt = \left[ -\frac{1}{2}t^2 + 6t \right]_{6}^{7} = 0.5
\]

#### 5. Line Segment (7 to 8)

\[
\int_{7}^{8} (8 - t) \, dt = \left[ 8t - \frac{1}{2}t^2 \right]_{7}^{8} = 0.5
\]

### Total Distance Traveled

\[
\text{Total Distance} = \frac{\pi}{2} + 3 + 6 + 0.5 + 0.5 = \frac{\pi}{2} + 10
\]

The total distance traveled by the ant is \(\frac{\pi}{2} + 10 \approx 11.57\).

Would you like more details or have any questions? Here are some related questions to consider:

1. How is the arc length of a semicircle calculated?
2. What is the significance of integrating the absolute value of a velocity function?
3. How does the velocity function relate to the ant's displacement?
4. What would be the effect on total distance if the velocity were negative in some intervals?
5. How do you compute the integral of a piecewise function?
6. What is the geometrical interpretation of integrating velocity?
7. How do changes in velocity affect the ant's total distance traveled?
8. How would the total distance change if the velocity graph had different shapes?

**Tip:** Understanding the relationship between a function and its integral is key in determining physical quantities like distance from velocity.

Explore how to calculate the total distance traveled by an ant using velocity analysis. Includes step-by-step integration of absolute value of velocity over time intervals, covering semicircles and linear segments. Ideal for advanced high school students and calculus enthusiasts.

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