Using summation notation to solve Riemann Sum problems

Question

I am learning how to write summation notation to solve Riemann Sum problems. I have a function

$f(x)=2x^3 -5x^2 +9x -5$ and I need to find the area under the curve with 8 intervals for $-4 \leq x \leq 12$ using midpoint Riemann Sum

I find the $\Delta x = 2$.

Then I use the midpoint formula $f(x_k) = a + \frac{1}{2}\Delta x + k \Delta x$ to find that $f(x_k) = -3 +2k$.

So I create the summation notation and plug the $f(x_k)$ value in place of $x$

$\sum_{k=0}^{n=7} (2(-3+2k)^3-5(-3+2k)^2+9(-3+2k)-5))2$

When I plug this into https://www.mathsisfun.com/numbers/sigma-calculator.html I get the answer 8,880 instead of the correct answer 7,648. Could some tell me where I am going wrong?

Answer 1

What you should have added is$$\sum_{k=0}^72\times f(-3+2k).$$Try it. You'll get $7\,648$.