Definite integral using Riemann sums?

In summary, the conversation discusses using Riemann sums to approximate the area under a curve and finding xi* using a formula. The "i" in the formula represents the number of the interval and can also be represented with specific values for a given problem.
  • #1
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I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for?

I was looking at a problem for f(x)=cos(x) and its bounded by x=0 and x=∏/2, with four subintervals. Also, the problem states to use the xi sample points as the midpoints. I understand how to do it, but I don't get what the "i" represents.

If this helps x1*=∏/16, x2*=3∏/16, x3*=5∏/16, and x4*=7∏/16

Thanks
 
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  • #2
i is just the number of the interval
x1*=∏/16, x2*=3∏/16, x3*=5∏/16, and x4*=7∏/16
could be written
xi*=(∏/16)(2i-1)
for i=1,2,3,4
 

1. What is a definite integral using Riemann sums?

A definite integral using Riemann sums is a method of approximating the area under a curve by dividing it into smaller rectangles and summing their areas. As the rectangles get smaller, the approximation becomes more accurate and approaches the actual value of the integral.

2. How is a Riemann sum calculated?

A Riemann sum is calculated by dividing the interval into smaller subintervals, determining the width of each subinterval, and then multiplying the width by the height of the function at a specific point within each subinterval. The sum of these values gives an approximation of the area under the curve.

3. What is the difference between a left Riemann sum and a right Riemann sum?

A left Riemann sum uses the left endpoint of each subinterval to determine the height of the rectangle, while a right Riemann sum uses the right endpoint. This can result in slightly different approximations of the integral, but as the number of subintervals increases, the difference becomes negligible.

4. What is the significance of the number of subintervals in a Riemann sum?

The number of subintervals directly affects the accuracy of the Riemann sum approximation. The more subintervals used, the closer the approximation will be to the actual value of the integral. However, using too many subintervals can also lead to computational errors and inefficiency.

5. How is a Riemann sum related to the definite integral?

A Riemann sum is a way to approximate the area under a curve, which is equivalent to finding the value of a definite integral. As the number of subintervals approaches infinity, the Riemann sum becomes a more accurate approximation of the definite integral.

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