use the limit process to find the area of the region between f(x) = x^2 + 2 a interval [0, 3]
To find the area of the region between the curve y = f(x) = x^2 + 2 and the x-axis over the interval [0, 3], you can use the limit process of Riemann sums. The Riemann sum is a method that breaks the area into infinitesimally small rectangles and sums up their areas.
Here's how to proceed:
1. Divide the interval [0, 3] into n equal subintervals. You can choose any positive integer n to control the accuracy of your approximation. Each subinterval will have a width of Δx = (3-0)/n.
2. Choose a representative point, xi, within each subinterval [xi-1, xi]. A common choice is to take xi as the right endpoint of each subinterval, making xi = xi-1 + Δx.
3. Calculate the height of the rectangle for each subinterval by evaluating f(xi) for each representative point. In this case, the height will be f(xi) = (xi)^2 + 2.
4. Calculate the area of each rectangle by multiplying the height by the width. The area of the ith rectangle is A_i = f(xi) * Δx.
5. Sum up the areas of all the rectangles to approximate the total area. The Riemann sum is given by the formula: R_n = Σ[A_i]. In this case, you need to sum up the areas of all the rectangles: R_n = Σ[(xi)^2 + 2 * Δx].
6. To find the exact area, take the limit of the Riemann sum as n approaches infinity. The exact area is given by the definite integral: Area = ∫[0, 3] (x^2 + 2) dx.
By evaluating this integral, you can find the exact area of the region between the curve y = x^2 + 2 and the x-axis over the interval [0, 3].