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11:50Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.
a. On what intervals is the object moving in the positive direction?
Determine whether the following statements are true and give an explanation or counterexample.
a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.
Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.
Theo: vT(t)=10, for t≥0
Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1
a. Graph the velocity function for both riders.
Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = ln x, for 1≤x≤4
For the given regions R₁ and R₂, complete the following steps.
a. Find the area of region R₁.
R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.
