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### Problem 7:

We are given the function:

\[
y = \frac{2x^2 - 4x}{(1 - 2x)(1 - x^2)}
\]

and asked to calculate the area bounded by the graph of this function, the \(x\)-axis, and the vertical lines \(x = -3\) and \(x = 2\), showing that this area is \(\ln(2.1)\).

### Approach:

To find the area under the curve, we will integrate the given function from \(x = -3\) to \(x = 2\), but since this involves a rational function, we'll simplify and compute the integral.

1. **Set up the integral:**
   The area under the curve can be found using:
   \[
   \text{Area} = \int_{-3}^{2} \frac{2x^2 - 4x}{(1 - 2x)(1 - x^2)} \, dx
   \]

   However, this integral might require breaking down into partial fractions or transformations to a logarithmic form.

---

### Problem 8:

(a) **Sketching to Show a Single Root**:

We are asked to show that the equation:

\[
e^{2x} + x - 2 = 0
\]

has only one root. To do this, we would plot the function \(f(x) = e^{2x} + x - 2\) and show that it only intersects the x-axis once. The graph of the exponential function \(e^{2x}\) grows rapidly and adding \(x - 2\) would shift and slightly modify the shape. Visually, you can confirm that it crosses the x-axis once.

(b) **Root Verification by Calculation**:

To verify that the root lies between \(x = 0\) and \(x = 0.5\), we check the sign of the function at both points:

1. Evaluate at \(x = 0\):
   \[
   f(0) = e^{2(0)} + 0 - 2 = 1 + 0 - 2 = -1
   \]

2. Evaluate at \(x = 0.5\):
   \[
   f(0.5) = e^{2(0.5)} + 0.5 - 2 = e^1 + 0.5 - 2 \approx 2.718 + 0.5 - 2 = 1.218
   \]

Since \(f(0)\) is negative and \(f(0.5)\) is positive, by the Intermediate Value Theorem, there is a root between \(x = 0\) and \(x = 0.5\).

---

Let me know if you want any details or clarifications. Here are 5 related questions for deeper understanding:

1. What are the key methods for solving integrals involving rational functions?
2. How does the Intermediate Value Theorem guarantee the existence of a root?
3. What other numerical methods can be used to find approximate roots of equations like \(e^{2x} + x - 2 = 0\)?
4. How does the behavior of exponential functions influence the number of roots of an equation?
5. How can we visualize the area under a curve using graphical tools?

**Tip**: For finding roots, using numerical methods such as the bisection method or Newton's method can give more precise results.

7. Show that the area bounded by the graph of y = (2x^2 - 4x) / ((1 - 2x)(1 - x^2)), the x-axis, and the line x = -3 and x = 2 is ln(2.1). 8. (a) By sketching a suitable pair of graphs show that the equation e^2x + x - 2 = 0 has only one root. (b) Verify by calculation that this root lies between x = 0 and x = 0.5.

This problem involves calculating the area under a curve using integration and showing that the exponential equation e^2x + x - 2 has only one root. The Intermediate Value Theorem is used to verify the root lies between 0 and 0.5. Suitable for advanced high school or university-level students.

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