To solve this problem, let’s break down the integral and the choices given.

The integral is: $\int_0^2 3 \, dx$

This integral represents the area under the function $f(x) = 3$ from $x = 0$ to $x = 2$.

Step-by-Step Solution:

1. Interpret the Function and Limits: The function $f(x) = 3$ is a constant function, meaning it produces a horizontal line at $y = 3$. When integrated from $x = 0$ to $x = 2$, we’re looking at a rectangle that spans horizontally from $x = 0$ to $x = 2$ and vertically from $y = 0$ to $y = 3$.

2. Calculate the Area Using Geometry: Since this region is a rectangle, we can find its area with the formula: $\text{Area} = \text{width} \times \text{height}$ where:

   • Width $= 2 - 0 = 2$
   • Height $= 3$

   Therefore: $\text{Area} = 2 \times 3 = 6$

Answer Selection:

Based on the sketches provided:

• We are looking for a rectangular region extending from $x = 0$ to $x = 2$ with a height of $y = 3$.

The correct answer is the second option (top-right image), which shows a rectangle with these dimensions.

Final Answer:

• The area calculated using the geometric formula is 6.

If you have more questions or need further details, feel free to ask!

Related Questions:

1. What is the integral of a constant function over any interval?
2. How can definite integrals represent areas under curves?
3. What would the integral $\int_0^3 4 \, dx$ represent in terms of area?
4. How does changing the upper limit of integration affect the area under a constant function?
5. How would the area change if $f(x) = 3x$ instead of $f(x) = 3$?

Tip:

When evaluating definite integrals of constant functions, you can quickly use the rectangle area formula: multiply the function value by the width of the interval.