# Unit 6 Integration and Accumulation of Change

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ENDURING UNDERSTANDING

CHA-4 Definite integrals allow us to solve problems involving the accumulation of change over an interval.

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 Topic Name Essential Knowledge 6.1 Exploring Accumulation of Change LEARNING OBJECTIVE CHA-4.A Interpret the meaning of areas associated with the graph of a rate CHA-4.A.1 The area of the region between the graph of a rate of change function and the x axis gives the accumulation of change. CHA-4.A.2 In some cases, accumulation of change can be evaluated by using geometry. CHA-4.A.3 If a rate of change is positive (negative) over an interval, then the accumulated change is positive (negative). CHA-4.A.4 The unit for the area of a region defined by rate of change is the unit for the rate of change multiplied by the unit for the independent variable.

Blog Posts

Integration Itinerary Order of topics in your integration unit.

The Old Pump  (11-30-2012) An exploration and introduction to Riemann sums and integration.

Flying into Integrationland  (12-3-2012) A continuation of the previous exploration.

Jobs, Jobs, Jobs (12-5-2012) Continuing the last two explorations, this time with real life data

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ENDURING UNDERSTANDING

LIM-5 Definite integrals can be approximated using geometric and numerical methods.

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 6.2 Approximating Areas with Riemann Sums LEARNING OBJECTIVE LIM-5.A Approximate a definite integral using geometric and numerical methods. LIM-5.A.1 Definite integrals can be approximated for functions that are represented graphically, numerically, analytically, and verbally. LIM-5.A.2 Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions. LIM-5.A.3 Definite integrals can be approximated using numerical methods, with or without technology. LIM-5.A.4 Depending on the behavior of a function, it may be possible to determine whether an approximation for a definite integral is an underestimate or overestimate for the value of the definite integral.

Blog Posts

Working Towards Riemann Sums

Variations on a Theme by ETS (6-14-2013) Adapting an exam problem on area and accumulation. . From the Good Question collection.

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 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation LEARNING OBJECTIVES LIM-5.B Interpret the limiting case of the Riemann sum as a definite integral. LIM-5.C Represent the limiting case of the Riemann sum as a definite integral LIM-5.B.1 The limit of an approximating Riemann sum can be interpreted as a definite integral. LIM-5.B.2 A Riemann sum, which requires a partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition. LIM-5.C.1 The definite integral of a continuous function  f over the interval [a, b], denoted by $\int_{a}^{b}{{f\left( x \right)dx}}$is the limit of Riemann sums as the widths of the subintervals approach 0. That is,  $\int_{a}^{b}{{f\left( x \right)dx}}=\underset{{\max \Delta {{x}_{1}}\to 0}}{\mathop{{\lim }}}\,\sum\limits_{{i=1}}^{n}{{f\left( {x_{i}^{*}} \right)}}\Delta {{x}_{i}}$where n is the number of subintervals, $\Delta {{x}_{i}}$is the width of the ith subinterval, and $x_{i}^{*}$ is a value in the ith subinterval. LIM-5.C.2 A definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

Blog Posts

Units The units of derivatives and integrals

Riemann Sums (12-12-2012) Left, right, midpoint, and Trapezoidal Riemann sums.

Variations on a Theme – 2 (6-28-2013) Practice with Riemann sums. From the Good Question collection.

Trapezoids – Ancient and Modern (2-7-2016)

The Definition of the Definite Integral (12-14-2012) And now we’re all set for …

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ENDURING UNDERSTANDING

FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.

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 6.4 The Fundamental theorem of Calculus and Accumulation Notation LEARNING OBJECTIVE FUN-5.A Represent accumulation functions using definite integrals. FUN-5.A.1 The definite integral can be used to define new functions. FUN-5.A.2 If f is a continuous function on an interval containing a, then $\frac{d}{{dx}}\left( {\int_{a}^{x}{{f\left( t \right)dt}}} \right)=f\left( x \right)$, where x is in the interval.

Blog Posts

Foreshadowing the FTC (12-15-2014) An example shows how the FTC works.

The Fundamental Theorem of Calculus (12-17-2012) Very important and fundamental. Relating derivatives and definite integrals.

More About the FTC (12-19-2012) What the FTC really means and why it’s important.

Good Question 11 – Riemann Reversed (11-29-2016) Given a Riemann sum find the associated function and its domain so you can find the integral. This is the reverse of the usual problem when one finds the Riemann sum first and is becoming a common question on the AP Calculus Exams. There are several examples and a discussion of the concern that the answer is never unique, which makes it a poor question.

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 6.5 Interpreting the Behavior of Accumulation Functions Involving Area LEARNING OBJECTIVE FUN-5.A Represent accumulation functions using definite integrals FUN-5.A.3 Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as $g\left( x \right)=\int_{a}^{x}{{f\left( t \right)dt}}$.

Blog Posts

Properties of Integrals (12-21-2018)

Units (1-26-2018) Determining the units of a definite integral

Logarithms (2-6-2013) Logarithms are defined by a definite integral.

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ENDURING UNDERSTANDING

FUN-6 Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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 6.6 Properties of Definite Integrals LEARNING OBJECTIVE FUN-6.A Calculate a definite integral using areas and properties of definite integrals. FUN-6.A.1 In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area. FUN-6.A.2 Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals. FUN-6.A.3 The definition of the definite integral may be extended to functions with removable or jump discontinuities.

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Properties of Integrals

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 6.7 The Fundamental Theorem of Calculus and Definite Integrals LEARNING OBJECTIVE FUN-6.B Evaluate definite integrals analytically using the Fundamental Theorem of Calculus. FUN-6.B.1 An antiderivative of a function f is a function g whose derivative is f. FUN-6.B.2 If a function f is continuous on an interval containing a, the function defined by $F\left( x \right)=\int_{a}^{x}{{f\left( t \right)dt}}$is an antiderivative of f for x  in the interval. FUN-6.B.3 If f is continuous on the interval [a, b] and F is an antiderivative of f, then $\int_{a}^{b}{{f\left( x \right)dx}}=F\left( b \right)-F\left( a \right).$

Blog Posts

Foreshadowing the FTC (12-15-2014) An example shows how the FTC works.

The Fundamental Theorem of Calculus (12-17-2012) Very important and fundamental. Relating derivatives and definite integrals.

More About the FTC (12-19-2012) What the FTC really means and why it’s important.

Good Question 11 – Riemann Reversed (11-29-2016) Given a Riemann sum find the associated function and its domain so you can find the integral. This is the reverse of the usual problem when one finds the Riemann sum first and is becoming a common question on the AP Calculus Exams. There are several examples and a discussion of the concern that the answer is never unique, which makes it a poor question.

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 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation LEARNING OBJECTIVE FUN-6.C Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives. FUN-6.C.1 $\int{{f\left( x \right)dx}}$ is an indefinite integral of the function f and can be expressed as $\int{{f\left( x \right)dx}}=F\left( x \right)+C$, where ${F}'\left( x \right)=f\left( x \right)$  and C is any constant. FUN-6.C.2 Differentiation rules provide the foundation for finding antiderivatives. FUN-6.C.3 Many functions do not have closed-form antiderivatives.

Blog Posts

Antidifferentiation  (11-28-2012)

Why Muss with the “+C”? But still don’t forget it.

Arbitrary Ranges (2-9-2014) Integrating inverse trigonometric functions.

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 6.9 Integration Using Substitution LEARNING OBJECTIVE FUN-6.D For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals. FUN-6.D.1 Substitution of variables is a technique for finding antiderivatives. FUN-6.D.2 For a definite integral, substitution of variables requires corresponding changes to the limits of integration.

Blog Posts

Antidifferentiation  (11-28-2012)

Why Muss with the “+C”? But still don’t forget it.

Arbitrary Ranges (2-9-2014) Integrating inverse trigonometric functions

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 6.10 Integrating Functions Using Long Division and Completing the Square LEARNING OBJECTIVE FUN-6.D For integrands requiring substitution or rearrangements into equivalent forms: (a) Determine indefinite integrals. (b) Evaluate definite integrals. FUN-6.D.3 Techniques for finding antiderivatives include rearrangements into equivalent forms, such as long division and completing the square.

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 6.11 Integration Using Integration by Parts  BC ONLY LEARNING OBJECTIVE FUN-6.E For integrands requiring integration by parts: (a) Determine indefinite integrals. BC ONLY (b) Evaluate definite integrals. BC ONLY FUN-6.E.1 Integration by parts is a technique for finding antiderivatives. BC ONLY

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Integration by Parts 1 (2-2-2013) Basics

Integration by Parts 2 (2-4-2013) The Tabular Method

Modified Tabular Integration (7-24-2013) A quicker way

Parts and More Parts (8-5-2016) Reduction formulas (Not tested on the AP Calculus exams)

Good Question 12 – Parts with a Constant (12-13-2016) How come you don’t need the “+C”?

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 6.12 Integration Using Linear Partial Fractions BC ONLY LEARNING OBJECTIVE FUN-6.F For integrands requiring integration by linear partial fractions: (a) Determine indefinite integrals. BC ONLY (b) Evaluate definite integrals. BC ONLY FUN-6.F.1 Some rational functions can be decomposed into sums of ratios of linear, nonrepeating factors to which basic integration techniques can be applied. BC ONLY

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 6.13 Evaluation Improper Integrals BC ONLY LEARNING OBJECTIVE LIM-6.A Evaluate an improper integral or determine that the integral diverges. BC ONLY LIM-6.A.1 An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of integration.  BC ONLY LIM-6.A.2 Improper integrals can be determined using limits of definite integrals. bc only. BC ONLY

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Improper Integrals and Proper Areas (1-25-2014)  A BC  topic.

Math vs. the “Real World” (2-2-2018) On the convergence of improper Integrals A BC topic

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 6.14 Selecting Techniques of Antidifferentiation none

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