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Figure 1. This is a classic exposition of the infinite series, the use of smaller and smaller areas which visually result in a finite area, but the key point with this depiction is often missed. It is not simply that these squares get closer and closer to filling the rectangle but that (1) the sequence of rectangles will never go outside the limiting rectangle, and (2) no smaller region will contain the sequence of rectangles. We know that these conditions together are satisfied if the ratio by which the rectangles' area decrease is less than 1. There are other conditions which assure this for us, though.
Figure 2. It helps to remember what an odd function looks like to get why function symmetry can be helpful with solving integrals.
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