Calc II Riemann Sum Help

[MirandaCaballero]

I have asked my teacher multiple times to explain this and I still have no idea how to solve. Help please!

Thank you!

 

[daon2]

I have asked my teacher multiple times to explain this and I still have no idea how to solve. Help please!
View attachment 5192
Thank you!

One of the definitions of the definite integral of a continuous function given in calculus is

\(\displaystyle \displaystyle \int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i)\cdot \Delta x\)

where \(\displaystyle \Delta x = \dfrac{b-a}{n}, x_i = a+i\cdot\Delta x\)

Now rewriting:

\(\displaystyle \displaystyle \lim_{n\to\infty} \ \sum_{i=1}^n 4\left( \dfrac{i}{n}\right)^3\cdot \dfrac{1}{n}\)

It may require some thought, but you should be able to figure out \(\displaystyle f(x),a,b\) from this.

 

[Ishuda]

I have asked my teacher multiple times to explain this and I still have no idea how to solve. Help please!
View attachment 5192
Thank you!

Do you know what a Riemann sum is? Suppose we have a function f(x) defined on an interval [0,a] and you divide the interval [0,a] into n pieces for our \(\displaystyle \Delta x\). That is
\(\displaystyle \Delta x = \frac{a}{n}\)
so that the ith interval starts at i * \(\displaystyle \Delta x\). That is
\(\displaystyle x_i = i\, *\, \Delta x = \frac{i\, a}{n}\)
The (partial) Riemann sum, R_n, which approximates the integral of f(x) from 0 to a, can now be written as
\(\displaystyle R_n\, =\, \underset{i=1}{\overset{n}{\Sigma}} f(x_i)\, \Delta x\, =\, \underset{i=1}{\overset{n}{\Sigma}} f(x_i)\, \frac{a}{n}\)

Choose the function f(x) and some value for a so that that Riemann sum looks like what you need.

 

[FUGU]

So that sum is equivalent with 4*(1³+....+n³)/n⁴. The sum has the following formula :[n(n+1)]² /2 .Doing some basic limit calculus it will result 2(n⁴+ 2*n³+n²)/n⁴​ . The limit is 2.