Area Under a Curve

[mskincaid]

The problem is "Use the limit process to find the area of the region bounded by the graphs of the given equations: g(x)=6x^2, y=0, x=1, x=4."
Could someone explain this process?

 

[daon]

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

Using n equal bases:

\(\displaystyle x_i = a+\frac{b-a}{n}i\)

\(\displaystyle \Delta x_i = \Delta x = \frac{b-a}{n}\)

Putting it all together:

\(\displaystyle \int_1^4 6x^2dx = \lim_{n \to \infty} \sum_{i=1}^n 6(1+\frac{4-1}{n}i)^2 \frac{4-1}{n}\)

\(\displaystyle = \lim_{n \to \infty} \sum_{i=1}^n 6(1+\frac{3}{n}i)^2 \frac{3}{n}\)

\(\displaystyle = \lim_{n \to \infty} \frac{18}{n}\sum_{i=1}^n 1+\frac{6}{n}i+\frac{9}{n^2}i^2\)

\(\displaystyle = \lim_{n \to \infty} \frac{18}{n}\Bug{[}\sum_{i=1}^n 1+\frac{6}{n}\sum_{i=1}^ni+\frac{9}{n^2}\sum_{i=1}^ni^2 \Big{]}\)

Can you finish from here?