91Ó°ÊÓ

In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$ f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \quad \text { on } \quad[0,4] $$

Suppose that \(\sum_{k=1}^{n} a_{k}=0\) and \(\sum_{k=1}^{n} b_{k}=1 .\) Find the values of $$ \begin{array}{ll}{\text { a. } \sum_{k=1}^{n} 8 a_{k}} & {\text { b. } \sum_{k=1}^{n} 250 b_{k}} \\ {\text { c. } \sum_{k=1}^{n}\left(a_{k}+1\right)} & {\text { d. } \sum_{k=1}^{n}\left(b_{k}-1\right)}\end{array} $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-4}^{0} \sqrt{16-x^{2}} d x $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\pi / 3}^{-\pi / 4}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) d t $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-2}^{1}|x| d x $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int 3 y \sqrt{7-3 y^{2}} d y $$

Evaluate the sums in Exercises \(19-28\). $$ \begin{array}{lll}{\text { a. } \sum_{k=1}^{10} k} & {\text { b. } \sum_{k=1}^{10} k^{2}} & {\text { c. } \sum_{k=1}^{10} k^{3}}\end{array} $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{\pi} 5(5-4 \cos t)^{1 / 4} \sin t d t $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{1}^{-1}(r+1)^{2} d r $$

Evaluate the sums in Exercises \(19-28\). $$ \begin{array}{llll}{\text { a. } \sum_{k=1}^{13} k} & {\text { b. } \sum_{k=1}^{13} k^{2}} & {\text { c. } \sum_{k=1}^{13} k^{3}}\end{array} $$