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05:362–3. Displacement, distance, and position Consider an object moving along a line with the following velocities and initial positions. Assume time t is measured in seconds and velocities have units of m/s.
d. Determine the position function s(t) using the Fundamental Theorem of Calculus (Theorem 6.1). Check your answer by finding the position function using the antiderivative method.
v(t) = 12t²-30t+12, for 0 ≤ t ≤ 3; s(0)=1
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
Variable gravity At Earth’s surface, the acceleration due to gravity is approximately g=9.8 m/s² (with local variations). However, the acceleration decreases with distance from the surface according to Newton’s law of gravitation. At a distance of y meters from Earth’s surface, the acceleration is given by a(y) = - g / (1+y/R)², where R=6.4×10⁶ m is the radius of Earth.
f. Graph ymax as a function of v0. What is the maximum height when v0=500 m/s,1500 m/s, and 5 km/s?
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Given only the velocity of an object moving on a line, it is possible to find its displacement, but not its position.
70–72. Variable density in one dimension Find the mass of the following thin bars.
A bar on the interval 0≤x≤6 with a density ρ(x) = {1 if 0 ≤ x < 2
2 if 2 ≤ x < 4
4 if 4 ≤ x ≤ 6
