The integral given is:

$\int_{0}^{5} 6 \, dx$

This represents the area of a region under the function $f(x) = 6$ from $x = 0$ to $x = 5$.

Steps to Sketch the Region:

1. Graph the Function: Plot the line $y = 6$, which is a horizontal line across the y-axis at $y = 6$.
2. Set the Limits: Mark the x-axis from $x = 0$ to $x = 5$.
3. Shade the Area: The area under the line $y = 6$ and between $x = 0$ and $x = 5$ represents the integral. This region forms a rectangle.

Dimensions and Area Calculation:

• The width of the rectangle (horizontal) is $5 - 0 = 5$.
• The height of the rectangle (vertical) is $6$.

Thus, the area is: $\text{Area} = \text{width} \times \text{height} = 5 \times 6 = 30$

Would you like a more detailed breakdown of the sketch or a deeper explanation?

Further Questions to Explore:

1. How would the area change if the integral limits were different?
2. What does this integral represent in practical terms?
3. How would you find the area if the function was not constant?
4. Can you find the area if the function was below the x-axis?
5. What is the general form of the integral for calculating the area under a curve?

Tip:

For constant functions, the integral over an interval simply represents the area of a rectangle with one side as the function value and the other as the interval length.