Use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. $\int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x$

The field strength H of a magnet of length 2L on a particle r units from the center of the magnet is

$$H=\frac{2 m L}{\left(r^2+L^2\right)^{3 / 2}}$$

where $\pm m$ are the poles of the magnet. Find the average field strength as the particle moves from 0 to R units from the center by evaluating the integral

$$\frac{1}{R} \int_0^R \frac{2 m L}{\left(r^2+L^2\right)^{3 / 2}} d r$$

A hydraulic cylinder on an industrial machine pushes a steel block a distance of x feet (0 ≤ x ≤ 5), where the variable force required is F(x) = 500x / √26-x² pounds. Find the work done in pushing the block the full 5 feet through the machine.

Use the weight 10-ton rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earth’s surface? (b) How far has the rocket traveled when half the total work has occurred?

For the years 1990 through 1997, the average size $S$ (in thousands of dollars) of ordinary life insurance policies in force in the United States is given in the table.

$$\begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Year } & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 \\ \hline S & 37.9 & 41.5 & 43.0 & 45.8 & 45.9 & 49.1 & 52.3 & 56.0 \\ \hline \end{array}$$

A model for these data is

$$S=\sqrt{1520.4+111.2 t+15.8 t^2}$$

where $t$ is the time in years, with $t=0$ corresponding to 1990 . Use a graphing utility to answer each of the following. (a) Graph the model for $0 \leq t \leq 7$. (b) Find the rate of increase in $S$ when $t=5$. (c ) Use the model and integration to predict the average value of $S$ for the years 2000 through 2002 .

Use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. $\int_{0}^{\pi / 2} e^{-2 x} \sin 3 x d x$

Use a computer algebra system to evaluate the definite integral. $\int_{0}^{\pi / 2}(1-\cos \theta)^{2} d \theta$

Use the weight of the rocket to answer the question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earth’s surface? (b) How far has the rocket traveled when half the total work has occurred? 5-ton rocket

Use a computer algebra system to evaluate the definite integral. $\int_{0}^{\pi / 4} \sin 3 \theta \sin 4 \theta d \theta$

Use L’Hôpital’s Rule to evaluate the limit. lim_(x→1^+) (2/ln x - 2/(x-1)

State the integration formula you would use to perform the integration. Explain why you chose that formula. Do not integrate. ∫ x(x² + 1)³ dx

L’Hopital’s Rule is used incorrectly. Describe the error. $\lim _{x \rightarrow 0} \frac{\sin \pi x-1}{x}=\lim _{x \rightarrow 0} \frac{\pi \cos \pi x}{1}=\pi$

Find the area of the surface formed by revolving the graph of y = 2e^-x on the interval [0, ∞) about the x-axis.