https://file.aiquickdraw.com/mathbot/file/1724743434915a9q4754r.jpg

Let's solve the two problems provided in the image by calculating the area of the region bounded by the given curves, the x-axis, and the vertical lines.

### **Problem 2a:**
**Given:**
- Curve: \( y = 2x \)
- X-axis: \( y = 0 \)
- Vertical lines: \( x = 1 \) and \( x = 4 \)

**1. Set up the integral:**
To find the area under the curve from \( x = 1 \) to \( x = 4 \), we calculate:
\[
\text{Area} = \int_{1}^{4} 2x \, dx
\]

**2. Compute the integral:**
\[
\int 2x \, dx = x^2
\]

**3. Evaluate from \( 1 \) to \( 4 \):**
\[
\text{Area} = \left[4^2\right] - \left[1^2\right] = 16 - 1 = 15
\]

So, the area of the region is **15 square units**.

---

### **Problem 2b:**
**Given:**
- Curve: \( y = 2x - 1 \)
- X-axis: \( y = 0 \)
- Vertical lines: \( x = 1 \) and \( x = 5 \)

**1. Find the x-intercept:**
First, let's find where the curve intersects the x-axis by setting \( y = 0 \):
\[
2x - 1 = 0 \implies x = \frac{1}{2}
\]

But since the bounds are between \( x = 1 \) and \( x = 5 \), we'll directly integrate between these limits.

**2. Set up the integral:**
\[
\text{Area} = \int_{1}^{5} (2x - 1) \, dx
\]

**3. Compute the integral:**
\[
\int (2x - 1) \, dx = x^2 - x
\]

**4. Evaluate from \( 1 \) to \( 5 \):**
\[
\text{Area} = \left[5^2 - 5\right] - \left[1^2 - 1\right] = \left[25 - 5\right] - \left[1 - 1\right] = 20 - 0 = 20
\]

So, the area of the region is **20 square units**.

---

### **Questions for Further Exploration:**
1. How do you find the area between two curves?
2. What is the significance of the limits of integration in calculating the area under a curve?
3. How do changes in the limits of integration affect the computed area?
4. How would you approach finding the area if the curve intersected the x-axis within the integration limits?
5. What is the geometric interpretation of a negative integral when calculating areas?

### **Tip:** 
Always visualize the problem first to ensure that you understand the region whose area you're trying to calculate. Drawing the curves and identifying the limits clearly can help avoid errors.

Explore how to calculate the area between curves using definite integrals. Includes step-by-step solutions for two examples: one with a linear function and another involving finding intercepts. Suitable for college-level students and anyone interested in integral calculus.

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