Study Guide for the Final exam

The integral

• Yes, you need to know the definition of Riemann sums and of integral. You also need to know the relation of the integral to the area between a graph and an axis, or between two graphs.
• You need to know also the statement of the Fundamental Theorem of Calculus. It is an important Theorem and we could ask you to state it. If we do, we need a complete statement, very much like the one in the notes. A formula with no explanation will give you very little partial credit.
• Learn well the difference between indefinite and definite integrals. Be sure you choose the right one when we ask for it. There will be heavy penalty for getting confused. Yes, you will find lots of integrals to calculate, be sure you know the ones in the box in page 96.
• The properties of the integral need to be known so you can use them if necessary (for example, the integral from a to b is the integral from a to c plus the integral from c to b. This can be used to break the calculation of the area between two graphs in several integrals according to how many times the graphs intersect). No proofs will be included in this last part of the exam.
• Differentiation of the integral when the variable appears in the limits of integration. This is very important, is in page 98. This is related to the Fundamental Theorem of Calculus (in fact, the rule for differentiating these integrals is often called the Second Fundamental Theorem of Calculus). Be sure to know the difference between the variable being at the top or at the bottom limit of integration.
• Yes, you will get some integrals where you will need to use u-substitution, it will show up on its own or as part of an application (or both).

Applications

• Areas between two graphs. It has appeared before, but it appears again. Find the intersection points and see which graph is larger. Calculate the areas accordingly breaking into several integrals if needed.
• Calculating the volume of a solid by slicing: you need to know that the volume is the integral of the area of a slice and, more importantly, you need to know how to find volumes using this principle!
• Volumes of revolution. They are the most important volumes in this section. But depending on how you rotate them the slices will be either disks or washers (flat donuts generated by rotating a segment). You might get a problem without indication to which case you have, so you need to learn to distinguish them and to calculate both. Tricky points: 1) which variable to choose as integration variable, 2) what are its integration bounds, and 3) how to write the area in terms of those variable. Learning this comes only with practice. Try to draw, it helps enormously and one gets better with practice. Don't be shy! we are bad enough drawers not to care too much how beautiful it looks.
• Calculation of volumes using cylindrical shells. Last method to learn, know the methos and how to apply it, and take special care not to mix it with the slicing method, they are quite different. Again, the trick is to figure out the radius and the altitude of the shells in terms of one of the variables, and to find the limits of integration for that variable. Practice!

More applications: Curve lengths, work, etc

• You need to know how to handle parametrized curves and what they are. This goes for both movements on a line, or on the plane. Learn how to use integrals and derivatives to relate displacement, velocity and acceleration (both ways, from displacement to acceleration, and back). Careful with the sign of the velocity or the acceleration, you need to know in which direction you have positive displacement (that would be the positive direction). Main example is free fall for displacement on a line, but any parametrized curve can appear on the plane. Learn the problems in page 118.
• You need to know how to find the speed of a parametrized curve on the plane, and its length (or total displacement).
• You need to know what the formula for work is and how to find it. Try to understand the examples.

Notation