Is the Calculation of Area Using Riemann Sum Correct?

[NIZBIT]

I was hoping someone could check my answer?

Use the limit of a Riemann sum to find the area of the region bounded by the graphs of y=2x^3+1, y=0, x=0, x=2.

Area=2

 

[Jameson]

The limit of the Riemann Sum is the integral. So evaluate $$\int_{0}^{2}2x^3+1dx$$. I don't get 2. How did you do your problem?

 

[benorin]

A useful formaula is

$$\int_{a}^{b}f(x) \, dx = \lim_{n\rightarrow\infty} \frac{b-a}{n}\sum_{k=1}^{n}f\left( a+\frac{b-a}{n}k\right)$$​

hence

$$\int_{0}^{2}(2x^3+1) \, dx = \lim_{n\rightarrow\infty} \frac{2}{n}\sum_{k=1}^{n}\left[ 2\left( \frac{2k}{n}\right) ^3 +1\right]$$​

 

[NIZBIT]

Now I am getting two answers-10 for the def integral and 8 for the sum.

 

[HallsofIvy]

10 is obviously the correct answer. How are you taking the limit, as n goes to infinity on
$$\lim_{n\rightarrow\infty} \frac{2}{n}\sum_{k=1}^{n}\left[ 2\left( \frac{2k}{n}\right) ^3 +1\right]$$
?

Do you know a formula for the sum of k³, k², and k that you are using?