Firstly for some useful notes and examples on Integration:

Integration notes University of Plymouth

University of Plymouth Notes

(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) = x4−3x3−4x2+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.
Definite Integral
Scrolling down the page we see that this query also returns the indefinite integral.
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.
Definite Integration

Definite Integration 2

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.

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