Paper , Order, or Assignment Requirements

Be very rigorous. Show all your work, and describe/explain every step.

1. The town of Integral, USA has a funny-looking hill. One year, the townspeople discovered that the inside of the hill was filled with chocolate, so they dug a hole straight down from the top to mine the chocolate out. Mathematicians have determined that the remaining volume of the hill can be found with calculus (yay!) by considering the region bound by y = 0, x = 2, and y = −x² + 4x + 46, and revolving it around the y−axis. Assume the units are in meters.
   • Include a graph of the region described before it is revolved around the y−axis. You may either draw it (neatly) yourself, or graph it on a computer system.
   • Include a graph of the scenario. You may either draw it (neatly) yourself, or graph it on a computer system.
   • Find the volume of the remaining portion of the hill. Describe how to slice the hill, and how to find the volume of a thin slice, before finding the total volume.
   • Assuming the top of the hill was perfectly flat before the chocolate was removed, Find the total original volume of the hill.
2. The great pyramid of Giza was approximately 146.5 meters tall when it was first built in 2560 BC (amazing, right?). The pyramid sits on a square base, with sides approximately 230.4 meters long.
   • Include a picture that represents the scenario. You may either draw it (neatly) yourself, or graph it on a computer system.
   • Find the volume of the great pyramid when it was first built. Describe how to slice the pyramid, and how to find the volume of a thin slice, before finding the total volume.