91Ó°ÊÓ

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{7} k(2 k+1) $$

Find the derivatives in Exercises \(27-30\) a. by evaluating the integral and differentiating the result. b. by differentiating the integral directly. $$ \frac{d}{d x} \int_{0}^{\sqrt{x}} \cos t d t $$

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{5} \frac{k^{3}}{225}+\left(\sum_{k=1}^{5} k\right)^{3} $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sin ^{5} \frac{x}{3} \cos \frac{x}{3} d x $$

Evaluate the sums in Exercises \(19-28\). $$ \left(\sum_{k=1}^{7} k\right)^{2}-\sum_{k=1}^{7} \frac{k^{3}}{4} $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \tan ^{7} \frac{x}{2} \sec ^{2} \frac{x}{2} d x $$

In Exercises \(29-32,\) graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) right- hand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$ f(x)=x^{2}-1, \quad[0,2] $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int r^{2}\left(\frac{r^{3}}{18}-1\right)^{5} d r $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int r^{4}\left(7-\frac{r^{5}}{10}\right)^{3} d r $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int x^{1 / 2} \sin \left(x^{3 / 2}+1\right) d x $$