Evaluating Riemann Integrals of f(x)=x^k where k>1 is an Integer

In summary, the conversation is about computing the Riemann integral of a function f:[0,1]->R where f(x)=x^k, with k>1 being an integer. The first step is to define a partition P_m of [0,1] using q_m= m^(-1/m) and evaluating L(f,P_m) and U(f,P_m). The next step is to show that the limits of L(f,P_m) and U(f,P_m) as n approaches infinity are both equal to 1/(k+1). It is then shown that f is integrable on [0,1] and the Riemann integral of f(x) on [0,1] is equal to
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To compute the Riemann integral of f:[0,1]->R given f(x)=x^k where k>1 is an integer
1. Let m>2 and define q_m= m^(-1/m) Let P_m be the partition of [0,1] given by P_m=(0< q_m^m < q_m^(m-1)< ...< q_m <1)
Explicitly evalute L(f,P_m) and U(f,P_m)
2. Show that lim n->inf. L(f,P_m)= 1/(k+1),and lim n->inf. U(f,P_m)= 1/(k+1)
3. Show that f is integrable on [0,1]
4. Show that integra[0,1] f(x) dx= 1/(k+1)
 
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Related to Evaluating Riemann Integrals of f(x)=x^k where k>1 is an Integer

1) What is a Riemann Integral?

A Riemann Integral is a mathematical concept used to calculate the area under a curve on a given interval. It is named after the German mathematician Bernhard Riemann.

2) What is the process for evaluating a Riemann Integral?

The process for evaluating a Riemann Integral involves dividing the interval into smaller subintervals, approximating the area under the curve with rectangles, and taking the limit as the number of subintervals approaches infinity.

3) How is the function f(x)=x^k where k>1 an example of a Riemann Integrable function?

The function f(x)=x^k where k>1 is an example of a Riemann Integrable function because it is continuous on the interval and can be approximated by a finite number of rectangles. This allows for the limit to be taken and the integral to be evaluated.

4) What is the significance of k>1 in the function f(x)=x^k when evaluating the Riemann Integral?

The condition of k>1 is significant because it ensures that the function is well-behaved and the area under the curve can be approximated accurately using rectangles. If k is a non-integer or less than 1, the function may not be integrable using Riemann's method.

5) Can Riemann Integrals be evaluated for functions other than polynomials?

Yes, Riemann Integrals can be evaluated for a wide range of functions, including trigonometric, exponential, and logarithmic functions. As long as the function is continuous on the interval, it can be approximated and integrated using Riemann's method.

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