In my maths course, my professor says we have to use the equation ∆x = (b − a)/n. This makes sense in explanation, but I cannot seem to figure out how we determine the value of n. In one of the examples that our professor gave us to work on gives us the following details:
the curve is y(x)=x^3, and we want to know the area under the curve from x=1 to x=3. it seems simple enough but I have no idea how to determine n, and my professor skimmed over it as if we all already knew.
If someone could explain this more clearly, that would be amazing!
The actual value of the area is $20$
Approximation becomes better increasing the number of rectangles, as you can see in the picture below. The price is the number of calculation which increases with $n$
$$Area_n=\sum_{k=1}^n \frac{2}{n}\,\left(1+\frac{2k}{n}\right)^3$$ take $n=4$
$$Area_4=\frac12\left[(1+1/2)^3+(1+1)^3+(1+3/2)^3+(1+2)^3\right]=27$$
Hope this helps $$..$$