Find the value of the definite integral using sums, not antiderivatives:
int= integral sign
int((x + 2)/(x^(3/2)),x= 16..25)dx
Please help... even just showing me how to get started would be greatly appreciated.
To find the value of the definite integral using sums, we can use a numerical method called the Riemann sum. The Riemann sum approximates the integral by dividing the interval into smaller subintervals and computing the sum of the areas of rectangles within those subintervals.
Here's how you can get started:
1. Determine the number of subintervals, denoted by n, that you want to use in the approximation. The larger the value of n, the more accurate the approximation will be. For now, let's start with n = 4.
2. Determine the width of each subinterval, often denoted by Δx. To do this, calculate the difference between the upper limit (25) and the lower limit (16), and then divide by the number of subintervals (n): Δx = (25 - 16) / n.
3. Write out the sum expression that represents the Riemann sum. For this problem, the sum expression will be:
sum(((x + 2)/(x^(3/2))) * Δx, i = 1 to n).
4. Substitute the appropriate values into the sum expression. In this case, replace Δx with the calculated value from step 2, and replace n with the chosen number of subintervals.
5. Evaluate the Riemann sum. Substitute the values of x for each subinterval into the function ((x + 2)/(x^(3/2))), multiply it by the width Δx, and sum up all the resulting products.
6. Repeat steps 1-5 with different values of n to see how the approximation improves with larger values of n. You can start with n = 4 and increase it to n = 10, 100, or even larger if desired.
By following these steps, you will be able to approximate the value of the definite integral using sums without relying on antiderivatives.