The purpose of finding the integral of (x^2) from 0 to 2 without using the fundamental theorem is to demonstrate an alternative method of calculating integrals using Riemann sums, which can be useful in certain situations where the fundamental theorem may not be applicable.
The Riemann sum method involves dividing the interval [0, 2] into smaller subintervals and approximating the area under the curve by summing the areas of rectangles formed by the function and the subintervals. As the number of subintervals increases, the approximation becomes more accurate and approaches the actual integral value.
First, the interval [0, 2] must be divided into equal subintervals of width Δx. Then, the midpoint of each subinterval is determined and used as the x-value for calculating the height of the rectangle. The area of each rectangle is then calculated by multiplying the height by Δx. Finally, the areas of all the rectangles are summed to approximate the integral value.
One limitation of using Riemann sums is that it can be time-consuming and tedious to calculate the areas of multiple rectangles, especially for functions that are not easily represented by simple geometric shapes. Additionally, the accuracy of the approximation depends on the number of subintervals used, so it may not always provide an exact result.
No, the Riemann sum method may not be applicable for all types of functions. It is most commonly used for continuous functions, and may not provide accurate results for functions with discontinuities or infinite discontinuities. In such cases, other methods of integration may be more suitable.