To solve for $\int_0^{10} f(x) \, dx$, we interpret the integral as the area under the graph of $f(x)$ between $x = 0$ and $x = 10$.

Step 1: Analyze the region

The graph shows a straight line segment starting at $(0, 5)$ and ending at $(10, 15)$. This is a triangle.

Step 2: Calculate the area of the triangle

• The base of the triangle is from $x = 0$ to $x = 10$, so the base is $10$ units.
• The height of the triangle is from $y = 5$ to $y = 15$, so the height is $10$ units.

The area of a triangle is given by: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting the values: $\text{Area} = \frac{1}{2} \times 10 \times 10 = 50$

Thus: $\int_0^{10} f(x) \, dx = 50$

Final Answer:

$\boxed{50}$

Would you like a breakdown of any specific part of this explanation?
Here are related questions to consider:

1. How does the slope of $f(x)$ affect the integral?
2. What happens if $f(x)$ crosses the $x$-axis in this interval?
3. Why is the integral interpreted as the area under the curve?
4. How would the result change if $f(x)$ was not linear?
5. Can we verify the calculation using geometry?

Tip: Always confirm the function's behavior (positive or negative) to decide if the integral represents area or needs adjustments for absolute values.