Problem 77
Explain why \(\left(1 / n^{3}\right) \sum_{i=1}^{n} i^{2}\) should be a good approximation to \(\int_{0}^{1} x^{2} d x\) for large \(n .\) Now calculate the summation expression for \(n=10\), and evaluate the integral by the Second Fundamental Theorem of Calculus. Compare their values.
Problem 79
Show that \(\frac{1}{2} x|x|\) is an antiderivative of \(|x|\), and use this fact to get a simple formula for \(\int_{a}^{b}|x| d x\).
Problem 82
Give an example to show that the accumulation function \(G(x)=\int_{a}^{x} f(x) d x\) can be continuous even if \(f\) is not continuous.
