A particle travels along a straight line with velocity $$ v(t) = 3e^{-t/2} \sin(2t) $$ meters per second. What is the total distance, in meters, traveled by the particle during the time interval $$ 0 \leq t \leq 2 $$ seconds?

Let R be the region bounded by the graphs of $$y=2x$$, $$y=4$$, and $$x=0$$. Which of the following gives the volume of the solid formed by revolving R about the line $$x=-1$$?

A particle moves along the $$x$$-axis. The velocity of the particle at time $$t$$ is given by $$v(t) = 2t - 6$$. If the position of the particle is $$x = 8$$ when $$t = 0$$, at what time $$t > 0$$ does the particle first return to its starting position?

What is the area of the region enclosed by the graphs of

The base of a solid is the region bounded by the graphs of $$y = \sin x$$ and $$y = 0$$ for $$0 \leq x \leq \pi$$. For the solid, each cross section perpendicular to the $$x$$-axis is an isosceles right triangle with legs along the $$y$$-axis. What is the volume of the solid?

The base of a solid is the region bounded by the graphs of $$y = \cos(x)$$ and $$y = -\cos(x)$$ for $$0 \leq x \leq \frac{\pi}{2}$$. For the solid, each cross section perpendicular to the $$x$$-axis is a rectangle whose height is half its width. What is the volume of the solid?

Let $$f$$ be the function defined by $$f(x)=\frac{2}{x+3}$$. What is the average value of $$f$$ on the interval $$[0, 3]$$?

A function $$d(t)$$ gives the rate of pollutant discharge from a factory, in kilograms per hour, where $$t$$ is measured in hours since midnight. Which of

The base of a solid is the region bounded by the $$x$$-axis and the graph of $$y = 4 - x^2$$. For the solid, each cross section perpendicular to the $$x$$-axis is a semicircle. What is the volume of the solid?

Find the area of the region bounded by the curves $$y = x^3 - x$$ and $$y = 3x$$.

Find the volume of the solid formed when the region bounded by $$y=x$$ and $$y=x^2$$ (with $$x$$ between $$0$$ and $$1$$) is revolved about the line $$y=-1$$ using the washer method.

A particle travels along the $$x$$-axis so that at time $$t \geq 0$$ its velocity is given by $$v(t) = t^2 - 6t + 8$$. What is the total distance the particle travels from $$t = 0$$ to $$t = 3$$?

A function $$v(t)$$ gives the rate of change of inventory in a warehouse, in items per hour, where $$t$$ is measured in hours since 8 A.M. Positive values

Given the position function $$s(t)=t^3-6*t^2+9*t+2$$, find the instantaneous velocity at $$t=3$$.

Refer to the graph above, which displays the function $$f(x)=3*x$$. Which of the following expressions correctly calculates the average value of $$f(x)$$ on the interval $$[0,8]$$?

A particle has acceleration $$a(t)=6*t+2$$. If its initial velocity is $$v(0)=5$$, find the velocity at $$t=3$$ seconds.

Which of the following equations is consistent with the Fundamental Theorem of Calculus relating an object's position $$s(t)$$ and its velocity $$v(t)$$ over the interval $$[a,b]$$? Do not use a calculator.

Consider the functions given above on the interval $$[0,2]$$. Their average values are computed using the formula $$\frac{1}{2}\int_0^2 f(x)\,dx$$. Rank these functions in increasing order of their average value.

Solve for $$r$$ in the equation $$\pi * r^2 = 25$$.