Firstly for some useful notes and examples on Integration:

**Plymouth University – Indefinite Integration****Plymouth University – Definite Integration**- Units 12-1 – 13.16 from
**AJ Hobson’s ‘Just the Maths’**offers very comprehensive coverage starting with the basics and going well beyond school Mathematics - Choose Integration from the
**MathCentre**topics **Helm Workbooks – section 13**

(These sources of Notes can all be found on the **Notes page**.)

You can use Desmos and WolframAlpha to check your work and see some excellent visual representations.

As an example, we can find the total area bounded by f(x) = x^{4}−3x^{3}−4x^{2}+12x, the x-axis, the line x=−1 and the line x=3.

We could use WolframAlpha for a quick check. The visual representation shows clearly that we are dealing with areas above and below the x-axis.

Scrolling down the page we see that this query also returns the indefinite integral.

For the total shaded area, we can change the limits of the query to evaluate each section.

See **Calculus & Analysis** for more examples of WolframAlpha queries.

Or we could turn to the excellent **Desmos** where we can very simply change the limits.

If you are unfamiliar with Integration with Desmos, turn to **Learn Desmos: Integrals**.

Note you can explore the **graph shown in the video**.

We could also look at this **introduction to Integration on GeoGebra**.