skyturnred said:Homework Statement
I have to approximate the integral of sin(sqrt(x)) from 0 to 40, with n=4, using the midpoint rule.
Homework Equations
The Attempt at a Solution
I found delta x to be 10, obviously, since I have to approximate from 0 to 40 using 4 large rectangles. I am having trouble finding f(.5(x(of i-1)+x(of i)))). I don't really even know where to start!
LCKurtz said:Don't get lost in the subscripts. You have four intervals. What are their midpoints? Just list their x values. They are the four points where you evaluate the function.
skyturnred said:Ok, so I dropped the formula and just tried to think it through myself. It makes sense in my mind that the first midpoint is f(1/2), the next is f(3/2), then f(5/2), etc. So in terms of i in sigma notation (if i=0 and the upper limit is 40), it should be f(i+(1/2))(10), right? But somehow I am still getting the wrong answer..
The midpoint rule is a numerical method for approximating the value of a definite integral. It involves dividing the interval of integration into smaller subintervals and using the midpoint of each subinterval to estimate the value of the integral.
To use the midpoint rule, you will need to first divide the interval of integration (0 to 40) into smaller subintervals. Then, for each subinterval, you will calculate the midpoint and evaluate the function (sin(sqrt(x))) at that point. Finally, you will multiply the sum of these values by the width of the subintervals and divide by the total number of subintervals.
The more subintervals you use, the more accurate your approximation will be. However, using too many subintervals can be computationally expensive. It is recommended to start with a smaller number of subintervals (e.g. 10-20) and increase gradually if needed.
The midpoint rule can be used for any function as long as it is continuous on the interval of integration. However, it may not always provide an accurate approximation, especially for functions with sharp curves or discontinuities.
You can compare your approximation to the exact value of the integral (if known) or use a smaller width of subintervals to see if the approximation improves. Additionally, you can also use other numerical methods (such as the trapezoid rule or Simpson's rule) to compare the results and determine the accuracy of your midpoint rule approximation.