Question

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Answer 1

To find the average you add something together and divide by the number of things, adding the top and bottom sides together and dividing by the number of sides (2) is averaging them. Just use equal intervals for an example

Answer 2

Sketch of the area using trapezoids:

The trapezoidal approximation \(T\) gives
\[\int_0^2x^2\,\mathrm{d}x\approx T= \frac{0^2+\left(\frac{1}{2}\right)^2}{2}+\frac{\left(\frac{1}{2}\right)^2+1^2}{2}+\frac{1^2+\left(\frac{3}{2}\right)^2}{2}+\frac{\left(\frac{3}{2}\right)^2+2^2}{2}\]where each term within the parens represents the area of each sub-interval trapezoid.

Regrouping the terms, you can write the sum as
\[\frac{0^2+\left(\frac{1}{2}\right)^2+1^2+\left(\frac{3}{2}\right)^2}{2}+\frac{\left(\frac{1}{2}\right)^2+1^2+\left(\frac{3}{2}\right)^2+2^2}{2}\]where the numerator of the first term is the lower approximation \(L\) and the second term is the upper approximation \(U\), so
\[T=\frac{L}{2}+\frac{U}{2}=\frac{L+U}{2}\]