Question

Use the limit process to find the area of the region between f(x) = x + 2 and the x-axis on the interval [0,3].

Answer 1

To find the area between f(x) = x + 2 and the x-axis on the interval [0, 3], use the limit process to sum the areas of infinitely many rectangles.

Step-by-step explanation:

To find the area of the region between the function f(x) = x + 2 and the x-axis on the interval [0, 3], we can use the limit process.

1. Start by setting up the limit as the width of the rectangle approaches zero:
   lim_{n \to \infty} {\sum_{i=0}^{n-1} {f(x_i) \cdot \Delta x}}
2. Next, partition the interval [0, 3] into smaller subintervals by taking the width of each rectangle to be \Delta x = \frac{3 - 0}{n}
3. Then, choose any point within each subinterval (x_i) and evaluate the function f(x) at that point
4. Finally, take the limit as the number of subintervals approaches infinity to get the area.

Plugging in the values, we have:
Area = lim_{n \to \infty} {\sum_{i=0}^{n-1} {(x_i + 2) \cdot \Delta x}}