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Session Review for Exam 4. Course Home Syllabus 1. Applications of Differentiation. The Definite Integral and its Applications. Techniques of Integration. Exploring the Infinite.

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Need help getting started? Don't show me this again Welcome!If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Skill Summary Legend Opens a modal. Finding the average value of a function on an interval.

Average value over a closed interval Opens a modal. Calculating average value of function over interval Opens a modal. Mean value theorem for integrals Opens a modal. Average value of a function Get 3 of 4 questions to level up! Connecting position, velocity, and acceleration functions using integrals.

Motion problems with integrals: displacement vs. Analyzing motion problems: position Opens a modal. Analyzing motion problems: total distance traveled Opens a modal. Motion problems with definite integrals Opens a modal. Worked example: motion problems with definite integrals Opens a modal. Average acceleration over interval Opens a modal. Analyzing motion problems integral calculus Get 3 of 4 questions to level up! Motion problems with integrals Get 3 of 4 questions to level up!

Using accumulation functions and definite integrals in applied contexts. Area under rate function gives the net change Opens a modal. Interpreting definite integral as net change Opens a modal. Worked examples: interpreting definite integrals in context Opens a modal. Analyzing problems involving definite integrals Opens a modal.

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Analyzing problems involving definite integrals Get 3 of 4 questions to level up! Problems involving definite integrals algebraic Get 3 of 4 questions to level up!

Quiz 1.Since we know how to get the area under a curve here in the Definite Integrals sectionwe can also get the area between two curves by subtracting the bottom curve from the top curve everywhere where the top curve is higher than the bottom curve.

And sometimes we have to divide up the integral if the functions cross over each other in the integration interval. Area of Region Between Two Curves.

Now that we know how to get areas under and between curves, we can use this method to get the volume of a three-dimensional solid, either with cross sections, or by rotating a curve around a given axis. When doing these problems, think of the bottom of the solid being flat on your horizontal paper, and the 3-D part of it coming up from the paper. Cross sections might be squares, rectangles, triangles, semi-circles, trapezoids, or other shapes.

Here are examples of volumes of cross sections between curves. Since we know now how to get the area of a region using integration, we can get the volume of a solid by rotating the area around a line, which results in a right cylinder, or disk.

So now we have two revolving solids and we basically subtract the area of the inner solid from the area of the outer one. Note that for this to work, the middle function must be completely inside or touching the outer function over the integration interval. Since I believe the shell method is no longer required the Calculus AP tests at least for the AB testI will not be providing examples and pictures of this method.

Please let me know if you want it discussed further. Click on Submit the arrow to the right of the problem to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

You can even get math worksheets. There is even a Mathway App for your mobile device. Skip to content. Area Between Curves Since we know how to get the area under a curve here in the Definite Integrals sectionwe can also get the area between two curves by subtracting the bottom curve from the top curve everywhere where the top curve is higher than the bottom curve.Skip to Main Content. District Home.

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### Free Calculus 1 Diagnostic Tests

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Continuity and Rational Functions Worksheet. Ch 1 AP Review Worksheet. The following worksheets are supplementary worksheets that you may choose to do Comprehensive Review Worksheet. Limits Worksheet - Graphical and Numerical. Derivative Wkst 3 - Sketching the derivative, Differentiability, and Velocity. Derivative Wkst 4 - Interpret the Derivative. Derivative Wkst 5 - Second Derivative. Ch 2 Practice Test. Chapter 3 Handouts:. Calc AB 3. Particles in Motion Wkst. Particles in Motion Answer Key.

Mean Value Theorem Worksheet Section 3. Mean Value Theorem Answer Key. L'hopital's Rule Worksheet Section 7. Lhopital Answer Key. Ch 3 Practice Test.

Ch 3 PT Answer Key.

**Calculus 2 - Integral Test For Convergence and Divergence of Series**

Sketching Antiderivatives Worksheet. Fundamental Theorem Worksheet - Lesson. Ch 4 Part 1 Practice Test.

Population Density Worksheet. Pop Density Answer Key. Videos going over and Worksheets:.

## Applications of integration

Calc AB Lesson.The last topic that we discussed in the previous section was the harmonic series. In that discussion we stated that the harmonic series was a divergent series. It is now time to prove that statement.

From the section on Improper Integrals we know that this is. So, just how does that help us to prove that the harmonic series diverges? Well, recall that we can always estimate the area by breaking up the interval into segments and then sketching in rectangles and using the sum of the area all of the rectangles as an estimate of the actual area.

The image below shows the first few rectangles for this area. Now note a couple of things about this approximation. First, each of the rectangles overestimates the actual area and secondly the formula for the area is exactly the harmonic series!

In other words, the harmonic series is in fact divergent. When discussing the Divergence Test we made the claim that. Again, from the Improper Integral section we know that. We will once again try to estimate the area under this curve. We will do this in an almost identical manner as the previous part with the exception that instead of using the left end points for the height of our rectangles we will use the right end points.

Here is a sketch of this case. This time, unlike the first case, the area will be an underestimation of the actual area and the estimation is not quite the series that we are working with. This means we can do the following. With the harmonic series this was all that we needed to say that the series was divergent.

Because the terms are all positive we know that the partial sums must be an increasing sequence. In other words. Therefore, the partial sums form an increasing and hence monotonic sequence. In the second section on Sequences we gave a theorem that stated that a bounded and monotonic sequence was guaranteed to be convergent.

This means that the sequence of partial sums is a convergent sequence. So, who cares right? Well recall that this means that the series must then also be convergent! So, once again we were able to relate a series to an improper integral that we could compute and the series and the integral had the same convergence. We went through a fair amount of work in both of these examples to determine the convergence of the two series.

The ideas in these two examples can be summarized in the following test. A formal proof of this test can be found at the end of this section.

There are a couple of things to note about the integral test. First, the lower limit on the improper integral must be the same value that starts the series. Second, the function does not actually need to be decreasing and positive everywhere in the interval. In other words, it is okay if the function and hence series terms increases or is negative for a while, but eventually the function series terms must decrease and be positive for all terms.Note that some sections will have more problems than others and some will have more or less of a variety of problems.

Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. Integration by Parts — In this section we will be looking at Integration by Parts. We also give a derivation of the integration by parts formula. Integrals Involving Trig Functions — In this section we look at integrals that involve trig functions.

In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions. Trig Substitutions — In this section we will look at integrals both indefinite and definite that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals.

Partial Fractions — In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions. Integrals Involving Roots — In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots. In some cases, manipulation of the quadratic needs to be done before we can do the integral. We will see several cases where this is needed in this section.

Integration Strategy — In this section we give a general set of guidelines for determining how to evaluate an integral. The guidelines give here involve a mix of both Calculus I and Calculus II techniques to be as general as possible.

Improper Integrals — In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite i. Determining if they have finite values will, in fact, be one of the major topics of this section.

Comparison Test for Improper Integrals — It will not always be possible to evaluate improper integrals and yet we still need to determine if they converge or diverge i.

So, in this section we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals — In this section we will look at several fairly simple methods of approximating the value of a definite integral.

It is not possible to evaluate every definite integral i. These methods allow us to at least get an approximate value which may be enough in a lot of cases. Practice Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode.

If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.Calculus I courses provide students with an in-depth introduction to the core concepts of limits, derivatives, and integrals, building on the preliminary understanding of these concepts that students gained in Pre-Calculus courses while preparing them for the more advanced material of Calculus II, Calculus II, and Differential Equations.

Calculus courses are often necessary for students to be able to tackle not only these higher-level mathematics courses, but advanced material in the sciences. If you are considering majoring in math, science, or any other quantitative field, taking Calculus before reaching college can be a real boon, as high school Calculus courses often take the material at a somewhat slower pace than collegiate courses, making sure that students fully understand each concept before moving on. The material taught in Calculus I courses can be broken down into three main categories: limits, derivatives, and integrals; however, most Calculus I courses begin with a review of the basic features of functions graphed on the coordinate plane, including continuity, asymptotes, and absolute and local extrema.

Students may be asked to find the slope of a line or slope at a point when reviewing these concepts. When delving into the concept of function limits, Calculus courses typically begin with the process of calculating and estimating simple limits and proceed to introduce concepts of asymptotes and continuity, calculating limits to infinity, and other complexities.

Discussing derivatives is a major part of every Calculus I course. It is crucial that students fully understand what derivatives represent as they progress in Calculus I, as they are soon asked to apply this knowledge by calculating derivatives at a point and of a function, as well as second derivatives.

They are also taught the Chain Rule. Students are also asked to graph derivatives and second derivatives, along with linear approximations of derivatives. As student knowledge of derivatives increases, Calculus I introduces the concepts of increasing and decreasing intervals, concavity and convexity, points of inflection, and slope fields. Students are also asked to make use of the Mean Value Theorem.

Specific derivatives, such as the derivatives of logarithms, exponents, sums, quotients, products, and trigonometric functions, are taught, as well as implicit differentiation. The last major topic of every Calculus I course is integrals. Integrals are introduced by talking about the definition of an integral, integral notation, definite integrals, and Riemann sums. After the concept of an integral is introduced in detail, students are taught the Fundamental Theorem of Calculus, how to take the integral of a function, and how to graph integrals.

Increasingly difficult problems are likely to appear, as students are asked to take the integral of more complex functions such as sums, quotients, and products, logarithms, exponents, and trigonometric functions.

Of all the math courses that students have the opportunity to take during high school, Calculus I has gained the reputation of being notoriously difficult. You receive detailed results after completing each one, as well as access to step-by-step explanations of how to arrive at the correct answer for each problem. If you find yourself needing to focus on problems that address one specific topic, such as the Fundamental Theorem of Calculus, you can do that, as problems are organized into Practice Tests by concept.

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