Integration and Accumulation of Change · ⁨التكامل وتراكم التغيير⁩

Enduring Understanding (CHA-4): Definite integrals allow us to solve problems involving the accumulation of change over an interval.

Learning Objective CHA-4.A: Interpret the meaning of areas associated with the graph of a rate of change in context.

• 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.

Enduring Understanding (LIM-5): Definite integrals can be approximated using geometric and numerical methods.

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.

Enduring Understanding (LIM-5): Definite integrals can be approximated using geometric and numerical methods.

Learning Objective LIM-5.B: Interpret 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.

Learning Objective LIM-5.C: Represent the limiting case of the Riemann sum as a definite integral.

• LIM-5.C.1 The definite integral of a continuous function $f$ over the interval $[a, b]$, denoted by $\int_{a}^{b} f(x)\,dx$, is the limit of Riemann sums as the widths of the subintervals approach 0. That is, $\int_{a}^{b} f(x)\,dx = \lim_{\max \Delta x_i \to 0} \sum_{i=1}^{n} f(x_i^*)\Delta x_i$, where $n$ is the number of subintervals, $\Delta x_i$ is the width of the $i$th subinterval, and $x_i^*$ is a value in the $i$th 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.

Enduring Understanding (FUN-5): The Fundamental Theorem of Calculus connects differentiation and integration.

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.
  • Illustrative examples for FUN-5.A.1: $f(x) = \int_{0}^{x} e^{-t^2}\,dt$.
• FUN-5.A.2 If $f$ is a continuous function on an interval containing $a$, then $\dfrac{d}{dx}\left( \int_{a}^{x} f(t)\,dt \right) = f(x)$, where $x$ is in the interval.

Enduring Understanding (FUN-5): The Fundamental Theorem of Calculus connects differentiation and integration.

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 $g(x) = \int_{a}^{x} f(t)\,dt$.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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 $F(x) = \int_{a}^{x} f(t)\,dt$ 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 $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Learning Objective FUN-6.C: Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.

• FUN-6.C.1 $\int f(x)\,dx$ is an indefinite integral of the function $f$ and can be expressed as $\int f(x)\,dx = F(x) + C$, where $F'(x) = f(x)$ 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.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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.

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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

Enduring Understanding (FUN-6): Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

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

Enduring Understanding (LIM-6): The use of limits allows us to show that the areas of unbounded regions may be finite.

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

This topic is intended to focus on the skill of selecting an appropriate procedure for antidifferentiation. Students should be given opportunities to practice when and how to apply all learning objectives relating to antidifferentiation.