Applications of integrals, part 1: Areas & Average Value

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…

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Starting Integration

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 sun) and the Fundamental theorem of Calculus. Theses posts help prepare students for Riemann sums. The Old Pump Where I start Integration Flying into…

The Definite Integral and the FTC

The Definition of the Definite Integral. The definition of the definite integrals is: If f is a function continuous on the closed interval [a, b], and   is a partition of that interval, and , then The left side of the definition is, of course, any Riemann sum for the function f on the interval [a,…

Antidifferentiation

We now turn to integration. The first thing to decide is when to teach antidifferentiation. Many books do this at the end of the last differentiation chapter or the first thing in the first integration chapter. Some teachers, myself included, prefer to wait until after presenting the Fundamental Theorem of Calculus. Still others wait until…

Other Derivative Applications

Some final applications of derivatives L’Hospital’s Rule  Locally Linear L’Hospital’s Demonstration of the proof L’Hospital Rules the Graph Good Question An AP Exam question that can be used to delve deeper into L’Hospital’s Rule (2008 AB 6) Related Rate problems Related Rates Problems 1   Related Rate Problems II Good Question 9  Baseball and Related Rates Painting a…

The Mean Value Theorem

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. Here are some previous post on the MVT: Fermat’s Penultimate Theorem   A lemma for Rolle’s Theorem:…

Extreme Values and Linear Motion

Two more applications of differentiation are finding extreme values and the analysis of linear motion. Extreme Values The Marble and the Vase Extremes without Calculus A Standard Problem Far Out! Linear Motion – Motion on a Line  Type 2 Problems Motion Problems: Same Thing, Different Context The Ubiquitous Particle Motion Problem  – a PowerPoint Presentation and…