1. As section 7.7 begins by saying, there are two instances when integrals need to be approximated rather than found using definite integrals - the second of which, when we're looking at real data rather than a function, has clear real-world applications. Any sort of numeric data collection that can't be described by a function (and where there is some sort of curvature) by definition does not have antiderivative, and therefore cannot be evaluated using a definite integral. In finance, for example, historical data for equity prices is clearly not described by a function. Nevertheless, an attempt to understand the area under the curve for a given security would require approximate integration.

3. Gabriel's Horn is a geometric object formed by the rotation of the function y=1/x about the x-axis (for x ?1). The figure is a paradox because while it has finite volume, it has infinite surface area. An analogy I came across to illustrate this was: the "horn" can be filled up with a finite amount of paint (cubic units), but that an infinite amount of paint would be required to cover its surface (square units). Integrals demonstrate this phenomenon:

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