Calculus 2. Calculate $\int_0^3 \left(x^2 +2\right) dx$ by the formal definition.

Question

Calculate by the formal definition of the definite Integral

$$ \int_0^3 \left(x^2 +2\right) dx.$$

I have

$\Delta x = \frac 3n,\ x_i^* = \frac {3i}n \\f(x_i) = (\frac {3i}n)^2 +2 $

Express $$ \sum_{k=1}^n f(x_i^*)\ \Delta x $$

in closed form, the answer will be in terms of $n$

The part i am stuck on is getting the constants out front of the $\sum$ sign.

I've plugged $x_i^* $ into $ f(x)$ and got $\frac {9i^2 + 2n}{n^2} $

but then i cant really figure out a way to isolate $i$ behind the $\sum$

Answer 1

You have $$ \sum_{i=1}^n f(x_i^*)\ \Delta x=\frac {27}{n^3} \sum_{i=1}^n i^2+ \frac {6}{n}\sum_{i=1}^n 1 $$ then use $$ \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} $$ which is proved here to get, as $n \to \infty$, $$ \sum_{i=1}^n f(x_i^*)\ \Delta x \to 15 $$