91Ó°ÊÓ

An object moves along a straight line with acceleration given by \(a(t)=\sin (\pi t)\). Assume that when \(t=0, s(t)=v(t)=0 .\) Find \(s(t), v(t),\) and the maximum speed of the object. Describe the motion of the object.

Does \(\int_{-\infty}^{\infty} \frac{x^{2}}{4+x^{6}} d x\) converge or diverge? If it converges, find the value.

A thin plate lies in the region contained by \(\sqrt{x}+\sqrt{y}=1\) and the axes in the first quadrant. Find the centroid.

Does \(\int_{-\infty}^{\infty} x d x\) converge or diverge? If it converges, find the value. Also find the Cauchy Principal Value, if it exists.

Use integration to compute the volume of a sphere of radius \(r\). You should of course get the well-known formula \(4 \pi r^{3} / 3\).

An object moves along a straight line with acceleration given by \(a(t)=1+\sin (\pi t)\). Assume that when \(t=0, s(t)=v(t)=0 .\) Find \(s(t)\) and \(v(t) .\)

Find the area bounded by the curves. \(y=\sin x \cos x\) and \(y=\sin x, 0 \leq x \leq \pi\)

An object moves along a straight line with acceleration given by \(a(t)=1-\sin (\pi t)\). Assume that when \(t=0, s(t)=v(t)=0 .\) Find \(s(t)\) and \(v(t) .\)

Does \(\int_{-\infty}^{\infty} \sin x d x\) converge or diverge? If it converges, find the value. Also find the Cauchy Principal Value, if it exists.

A hemispheric bowl of radius \(r\) contains water to a depth \(h\). Find the volume of water in the bowl.