[Calculator] The average value of the function \(f(x) = x^2 \sin x\) on the interval \([2, 4]\)

[Calculator] Find the distance traveled in the first four seconds, for a particle whose velocity is given by \(v(t) = 7 \sin(t)\), where \(t\) represents time, in seconds.

The volume generated by revolving about the x-axis the region below the curve \(y = x^3\), above the x-axis, and between \(x = 0\) and \(x = 1\) is

If \(f\) is a continuous function, and \(F'(x) = f(x)\) for all real numbers \(x\), then \(\displaystyle\int_{1}^{3} f(2 x) d x =\)

[Calculator] Let \(g\) be the function given by \(g(x) = \displaystyle\int_{0}^{x} \sin(t^2) d t\) for \(-1 \leq x \leq 3\). On which of the following intervals is \(g\) decreasing?

If the region enclosed by the y-axis, the curve \(y = 4 \sqrt{x}\), and the line \(y = 8\) is revolved about the x-axis, the volume of the solid generated is

[Calculator] Find the length of the curve \(y = x^{\dfrac{3}{2}}\) from \(x = 1\) to \(x = 2\)

What is the average value of \(y = \sin 2 x\) over the interval \([\dfrac{\pi}{4}, \dfrac{\pi}{3}]\)?

Which of the following integrals correctly corresponds to the area of the shaded region between \(f(x) = 1 + x^2\) and \(g(x) = 5\), in the first quadrant from \(x = 1\) to \(x = 2\)?

A particle's position is given by \(s(t) = \sin t + 2 \cos t + \dfrac{t}{\pi} + 2\). The average velocity of the particle over \([0, 2 \pi]\)

A solid is generated when the region in the first quadrant enclosed by the graph of \(y = (x^2 + 1)^3\), the line \(x = 1\), the x-axis, and the y-axis is revolved about the x-axis. Its volume is found by evaluating which of the following integrals?

\(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos x d x =\)

\(\dfrac{d}{d x} \displaystyle\int_{0}^{x} \sin(t) d t =\)