To evaluate the integral $\int_{2}^{6} (3 + x) \, dx$ using geometry, let’s interpret the expression $3 + x$ as defining a function. This represents a line with a slope of $1$ and a y-intercept of $3$.

Step-by-Step Solution

1. Set up the integral: $\int_{2}^{6} (3 + x) \, dx$

2. Interpret the Region: The function $f(x) = 3 + x$ is a linear function on the interval $[2, 6]$. The area under this line from $x = 2$ to $x = 6$ can be found by calculating the area of the trapezoid formed between $x = 2$ and $x = 6$.

3. Determine Function Values:

   • At $x = 2$: $f(2) = 3 + 2 = 5$
   • At $x = 6$: $f(6) = 3 + 6 = 9$

4. Calculate the Area of the Trapezoid: For a trapezoid with parallel sides of lengths $a$ and $b$, and height $h$, the area $A$ is: $A = \frac{1}{2} \times (a + b) \times h$

   • Here, $a = f(2) = 5$
   • $b = f(6) = 9$
   • $h = 6 - 2 = 4$

   Plugging in these values: $A = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28$

Final Answer

The exact value of the integral is: $\int_{2}^{6} (3 + x) \, dx = 28$

Would you like further details on any step or have additional questions?

Related Questions

1. How can we verify this result by computing the integral algebraically?
2. What is the geometric interpretation of integrals involving linear functions?
3. How does the trapezoid formula apply to different types of functions?
4. Can we solve this problem using definite integral properties?
5. How would the result change if the limits of integration were altered?

Tip: For linear functions, the area under the curve on a specified interval can often be found using basic geometric shapes like trapezoids or triangles.