Exercises

1. Evaluate each of the following integrals.

   \(\displaystyle\int \left(t^2+\sin\,3t\right)\,dt\)	\(\displaystyle\int\;\frac{7}{\sqrt{s+1}}\,ds\)	\(\displaystyle\int_0^{\,\pi/2} \cos\,2x \,dx\)
   \(\displaystyle\int_0^{\,2}\,e^{-0.3t}\,dt\)	\(\displaystyle\int\,\left(1+x^2\right)^2\,dx\)	\(\displaystyle\int_0^{\,\pi/2} \,\sqrt{u}\;du\)

2. Find the center of mass of each of the following regions. Calculate each integral by hand, and check your work with the Definite Integral tool.
   1. The region bounded above by the curve \(y=2-x^2\) and below by the \(x\)-axis.
   2. The region bounded above by the line \(y=x\) and below by the curve \(y=x^2\).
   3. The region in the first quadrant bounded above by the curve \(y=2x^4\), below by the \(x\)-axis, and on the right by \(x=2\).
   4. The region in the first quadrant bounded above by the curve \(y=1+x^2\), below by the \(x\)-axis, on the left by \(x=0\), and on the right by \(x=1\).
   5. The region in the first quadrant bounded above by the curve \(y=\sqrt{x}\) and below by \(y=x\).
   6. The region in the first quadrant bounded above by the curve \(y=x^3\), below by \(y=x\), on the left by \(x=1\), and on the right by \(x=2\).
3. Find the center of mass of the half-disk under the graph of \(x^2+y^2=1\) and above the \(x\)-axis, following these steps:
   1. First, without any calculation, what is \(\bar{x}\)? What is the area?
   2. Now set up an integral for \(M_x\) and evaluate it.
   3. What is \(\bar{y}\)?
4. Find the center of mass of the quarter-circular region under the graph of \(x^2+y^2=1\) and in the first quadrant. (Hint: \(\bar{x}\) must be the same as \(\bar{y}\) - why? Thus you need at most one moment calculation. Use as much of the previous exercise as you can - you may be able to solve this problem without any calculation at all.)