One of the major applications of integration is to find the volumes of various solid figures. Volume of Solids with Regular Cross-sections This is where to start with volume problems. After all, solids of revolution are just a special case of solids with regular cross-sections. Volumes of Revolution Subtract the Hole from the Whole and…
So, you’ve finally proven the Fundamental Theorem of Calculus and have written on the board: And the students ask, “What good is that?” and “When are we ever going to use that?” Here’s your answer. There are two very important uses of this theorem. First, in words the theorem says that “the integral of a…
Usually the first application of integration is to find the area bounded by a function and the x-axis, followed by finding the area between two functions. We begin with these problems First some calculator hints Graphing Integrals using a graphing calculator to graph functions defined by integrals Graphing Calculator Use and Definition Integrals – Exam…
Why “scaling” is necessary No teacher can make two tests on the same topics equal in difficulty. No two teachers, even if they collaborate, can make two tests on the same topic equal in difficulty. No two teachers in different schools, districts, or states can make two tests on the same subject equal in difficulty.…
I got to thinking about grading the other day after seeing a question on Facebook. We’ll get to that question in a minute, but first I want to try to outline a grading scheme I used towards the end of my teaching career. It is based on how free-response question on AP Calculus exams are…
I had an email last week from a teacher asking, how come I can use a substitution to find a power series for , and for , but not for ? The answer is that you can. Substituting (2x) in to the cosine’s series give you a Taylor series centered at x = 0, a…
The series of post leads up to the Fundamental theorem of Calculus (FTC). Obviously, a very important destination. Working Towards Riemann Sums Definition of the Definite Integral and the FTC – a more exact demonstration from last Friday’s post and The Fundamental Theorem of Calculus – an older demonstration More about the FTC The derivative of a function…
Happy Thanksgiving! I’ve been working this month on making things at the Teaching Calculus blog easier to find. There are about 360 posts and it’s getting difficult for me to find things. Here is the new line up on the black navigation menu at the top of each page. Click each to see more. HOME…
Behind every definite integral is a Riemann sums. Students need to know about Riemann sums so that they can understand definite integrals (a shorthand notation for the limit if a Riemann sum) and the Fundamental theorem of Calculus. Theses posts help prepare students for Riemann sums. Integration Itinerary Some thoughts on the order of topics…
Integration – DON’T PANIC As I’ve mentioned before, I try to stay a few weeks ahead of where I figure you are in the curriculum. So here. early in November, I start with integration. You probably don’t start integration until after Thanksgiving in early December. That’s about the midpoint of the year. Don’t wait too…
Related Rate Questions Related Rate questions are an application of derivative. If two or more quantities help model the same situation, then their derivatives are related and may be used to examine their rates of change. these are called related rate problems. They appear on the AP Calculus exams usually as part of a free-response or…
Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year. See last Fridays post Foreshadowing the MVT for an a series of problems that will get…
