1a)For area between the curves, you just need to examine the curves. You basically wanna see which curve is the "upper" curve, or which one is on top, then the formula is very simple
being the "upper curve" and being the "lower curve"
1b) For volume, it depends on what exactly you're doing (what is access or rotation, and what method you are most comfortable with)
This situation would be most simple to use the "shells" method, where the formula is much the same as the one above, I will use the same notation of as upper function and as the lower function.
The formula then becomes
You can visualize this integral as the "radius" of 2 circles. Since you need to rotate around y=4, you just have to add in a factor of 4 to this radius so it becomes
1c) For this last part, you are told so you can just use the rules you normally know for finding maxima and minima by simply taking the derivative. Keep in mind, however, that you must also test the end points of the interval.
2a) This part is quite simple. Simply look at the chart, and see that
2b) This part is also quite simple. You must use the formula for a Right Hand Reimann Sum. The formula is quite simple and works out to be You need 5 subintervals, so your will change, as you input values, but the formula remains the same. You just need to multiply each by their respective
2c) This part is simply analysis. If you are given the rate of change of , the to integrate said function over an interval would give you the total change in that occurs over the interval, and the unit would, obviously, be in feet.
Hope this helps! Good luck on the exam!