91Ó°ÊÓ

Express the sums in closed form. $$ \sum_{k=1}^{n-1} \frac{k^{3}}{n^{2}} $$

True-False Determine whether the statement is true or false. Explain your answer. (Assume that \(f\) and \(g\) denote continuous functions on an interval \([a, b]\) and that \(f_{\mathrm{ave}}\) and \(g_{\mathrm{ave}}\) denote the respective average values of \(f \text { and } g \text { on }[a, b] .)\) If \(g_{\text {ave }}

Determine whether the statement is true or false. Explain your answer. If \(A(n)\) denotes the area of a regular \(n\) -sided polygon inscribed in a circle of radius \(2,\) then \(\lim _{n \rightarrow+\infty} A(n)=2 \pi\).

Evaluate the integrals using appropriate substitutions. $$ \int \sec 4 x \tan 4 x d x $$

Evaluate the integral and check your answer by differentiating. $$ \int \frac{x^{5}+2 x^{2}-1}{x^{4}} d x $$

Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. $$ \int_{-\pi / 2}^{\pi / 2} \sin \theta d \theta $$

Evaluate the integral and check your answer by differentiating. $$ \int \frac{1-2 t^{3}}{t^{3}} d t $$

Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. $$ \int_{0}^{\pi / 4} \sec ^{2} \theta d \theta $$