Let's analyze the given problem. The task is to evaluate the integral of the function $H(x)$ from $x = -7.4$ to $x = 10$, based on the graph provided.

To evaluate this integral, we need to compute the area under the curve of the function $H(x)$ between the limits of integration, $x = -7.4$ and $x = 10$.

From the graph:

1. Identify the shape of the curve, which appears to be piecewise linear (possibly a V-shape).
2. Break the graph into sections where the function changes direction or has a constant slope.
3. Calculate the area of each section (e.g., triangles, rectangles) under the curve.
4. Sum the areas of all sections to find the total area under the curve.

I'll now proceed with these steps based on the graph provided.To solve this problem, let's break the curve into manageable parts based on the visual inspection of the graph.

1. The graph of $H(x)$ appears to be a piecewise linear function:

   • One part of the graph is a straight line from $x = -7.4$ to where it meets the x-axis around $x = -4$.
   • Another part of the graph is a straight line from $x = -4$ to $x = 0$.
   • Then, from $x = 0$, it continues to rise steeply, with the function having a slope between $x = 0$ and $x = 10$.

2. Break the problem into areas:

   • From $x = -7.4$ to $x = -4$, the area is a triangle (since the graph is a straight line).
   • From $x = -4$ to $x = 0$, the area is another triangle.
   • From $x = 0$ to $x = 10$, the area is a larger triangle.

Steps to calculate the areas:

• Area from $x = -7.4$ to $x = -4$:

  • Base = $-4 - (-7.4) = 3.4$
  • Height = $5$ (based on the graph, as the line touches 5 on the y-axis at $x = -4$).
  • Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3.4 \times 5 = 8.5$

• Area from $x = -4$ to $x = 0$:

  • Base = $0 - (-4) = 4$
  • Height = $4$ (the function reaches 4 at $x = 0$).
  • Area of triangle = $\frac{1}{2} \times 4 \times 4 = 8$

• Area from $x = 0$ to $x = 10$:

  • Base = $10 - 0 = 10$
  • Height = $10$ (the function reaches 10 at $x = 10$).
  • Area of triangle = $\frac{1}{2} \times 10 \times 10 = 50$

Total Area:

The total area under the curve is the sum of the three areas: $\text{Total Area} = 8.5 + 8 + 50 = 66.5$

Thus, the integral evaluates to $\int_{-7.4}^{10} H(x) \, dx = 66.5$.

Let me know if you need further details or explanations!