Exercises

1. Evaluate each of the following indefinite integrals.

   \(\displaystyle\int \sin\,3x\,dx\)	\(\displaystyle\int\, e^{3t+1} \,dt\)	\(\displaystyle\int \,\frac{du}{1+3u}\)
   \(\displaystyle\int \,(1+x)^3\,dx\)	\(\displaystyle\int\,\frac{dt}{t(1+3t)}\)	\(\displaystyle\int \,\sqrt{u}\,du\)
   \(\displaystyle\int \,\sqrt{1+x}\,dx\)	\(\displaystyle\int \,\cos\left(\frac{t}{3}\right)\,dt\)	\(\displaystyle\int \,\sin(u+\pi)\,du\)

2. Evaluate each of the following definite integrals. Use the Definite Integral tool to check your work. (Note that the tool expects a function of \(x\).)

   \(\displaystyle\int_0^{\,\pi/3} \sin\,3x\,dx\)	\(\displaystyle\int_0^{\,1}\, e^{3t+1} \,dt\)	\(\displaystyle\int_0^{\,1} \,\frac{du}{1+3u}\)
   \(\displaystyle\int_1^{\,2}\,(1+x)^3\,dx\)	\(\displaystyle\int_1^{\,2}\,\frac{dt}{t(1+3t)}\)	\(\displaystyle\int_0^{\,4}\, \sqrt{u}\;du\)
   \(\displaystyle\int_{-1}^{\,3}\, \sqrt{1+x}\,dx\)	\(\displaystyle\int_0^{\,\pi}\, \cos\left(\frac{t}{3}\right)\,dt\)	\(\displaystyle\int_0^{\,\pi/2}\, \sin(u+\pi)\,du\)
3. Find the center of mass of each of the following systems of point masses. The masses are in grams and the distances in centimeters.

   a.    Mass \(x\)-coordinate \(15\) \(20\) \(10\) \(35\) \(20\) \(50\)	b.    Mass \(x\)-coordinate \(20\) \(-25\) \(10\) \(-12\) \(25\) \(13\)
   c.    Mass \(x\)-coordinate \(23\) \(-27\) \(18\) \(-12\) \(45\) \(10\) \(70\) \(28\) \(33\) \(35\)	d.    Mass \(x\)-coordinate \(100\) \(10\) \(150\) \(30\) \(50\) \(40\) \(75\) \(70\)

4. A stiff horizontal rod of negligible mass has 1-ounce fishing weights suspended at 6, 14, and 20 inches from the left end. In addition, 3-ounce fishing weights are suspended at 2 and 18 inches from the left end. Where is the balance point of this system?
5. A stiff horizontal rod of negligible mass has 1-ounce fishing weights suspended at 5, 15, and 25 inches from the right end. In addition, 2-ounce fishing weights are suspended at 10 and 20 inches from the right end. Where is the balance point of this system?
6. The following weights are suspended from a 1-meter rod of negligible mass:
   
   
   • 50 grams at 10 centimeters from the left end
   • 60 grams at 40 centimeters from the left end
   • 20 grams at 70 centimeters from the left end

   Where should a 60-gram weight be suspended to place the balance point in the middle of the rod?

7. The following weights are suspended from a 1-meter rod of negligible mass:
   
   
   • 50 grams at 10 centimeters from the left end
   • 60 grams at 40 centimeters from the left end
   • 20 grams at 70 centimeters from the left end

   Where should a 60-gram weight be suspended to place the balance point in the middle of the rod?