Chapter 5

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

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{\sqrt[3]{\pi^{2}}} \sqrt{\theta} \cos ^{2}\left(\theta^{3 / 2}\right) d \theta $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \cos (3 z+4) d z $$

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{6}\left(3-k^{2}\right) $$

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

In Exercises \(23-26,\) use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=\sin x \quad \text { on } \quad[0, \pi] $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{-1}^{-1 / 2} t^{-2} \sin ^{2}\left(1+\frac{1}{t}\right) d t $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sin (8 z-5) d z $$

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{6}\left(k^{2}-5\right) $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{9}^{4} \frac{1-\sqrt{u}}{\sqrt{u}} d u $$

Use areas to evaluate the integrals in Exercises \(23-26\) $$ \int_{0}^{b} 4 x d x, \quad b > 0 $$

In Exercises \(23-26,\) use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi] $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sec ^{2}(3 x+2) d x $$

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

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-4}^{4}|x| d x $$

Use areas to evaluate the integrals in Exercises \(23-26\) $$ \int_{a}^{b} 2 s d s, \quad 0 < a < b $$

In Exercises \(23-26,\) use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] $$

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

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

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi} \frac{1}{2}(\cos x+|\cos x|) d x $$

In Exercises \(23-26,\) use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$ f(x)=x \sin ^{2} \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] $$

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

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

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

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

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

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

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.) $$ f(x)=\sin x, \quad[-\pi, \pi] $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{x} \sqrt{1+t^{2}} d t $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int x^{1 / 3} \sin \left(x^{4 / 3}-8\right) d x $$

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.) $$ f(x)=\sin x+1, \quad[-\pi, \pi] $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{1}^{x} \frac{1}{t} d t, \quad x>0 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sec \left(v+\frac{\pi}{2}\right) \tan \left(v+\frac{\pi}{2}\right) d v $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \csc \left(\frac{v-\pi}{2}\right) \cot \left(\frac{v-\pi}{2}\right) d v $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{x^{2}} \cos \sqrt{t} d t $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{\sin (2 t+1)}{\cos ^{2}(2 t+1)} d t $$

Find \(d y / d x\) in Exercises \(31-36\) $$ y=\int_{0}^{\sin x} \frac{d t}{\sqrt{1-t^{2}}}, \quad|x|<\frac{\pi}{2} $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{6 \cos t}{(2+\sin t)^{3}} d t $$

For the functions in Exercises \(35-40\) find a formula for the upper sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals. Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). $$ f(x)=2 x \text { over the interval }[0,3] $$

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

In Exercises \(37-42,\) find the total area between the region and the \(x\) -axis. $$ y=-x^{2}-2 x, \quad-3 \leq x \leq 2 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{\sec z \tan z}{\sqrt{\sec z}} d z $$

For the functions in Exercises \(35-40\) find a formula for the upper sum obtained by dividing the interval \([a, b]\) into \(n\) equal subintervals. Then take a limit of these sums as \(n \rightarrow \infty\) to calculate the area under the curve over \([a, b]\). $$ f(x)=3 x^{2} \text { over the interval }[0,1] $$

In Exercises \(37-42,\) find the total area between the region and the \(x\) -axis. $$ y=3 x^{2}-3, \quad-2 \leq x \leq 2 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{1}{t^{2}} \cos \left(\frac{1}{t}-1\right) d t $$

In Exercises \(37-42,\) find the total area between the region and the \(x\) -axis. $$ y=x^{3}-3 x^{2}+2 x, \quad 0 \leq x \leq 2 $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{1}{\sqrt{t}} \cos (\sqrt{t}+3) d t $$