Unit 8: Applications of Integration
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- 2020
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Topic 8.1
8.1 Finding the Average Value of a Function on an Interval
Objectives in this topic
CHA-4.B—Determine the average value of a function using definite integrals
- CHA-4.B Determine the average value of a function using definite integrals.
- CHA-4.B.1 The average value of a continuous function f over an interval [a, b] is (1/(b − a))∫ₐᵇ f(x)dx.
- Enduring understanding CHA-4: Definite integrals allow us to solve problems involving the accumulation of change over an interval.
Topic 8.2
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
Objectives in this topic
CHA-4.C—Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion
- CHA-4.C Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion.
- CHA-4.C.1 For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle’s displacement over the interval of time, and the definite integral of speed represents the particle’s total distance traveled over the interval of time.
- Enduring understanding CHA-4: Definite integrals allow us to solve problems involving the accumulation of change over an interval.
Topic 8.3
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Objectives in this topic
CHA-4.D—Interpret the meaning of a definite integral in accumulation problems
- CHA-4.D Interpret the meaning of a definite integral in accumulation problems.
- CHA-4.D.1 A function defined as an integral represents an accumulation of a rate of change.
- CHA-4.D.2 The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.
- Enduring understanding CHA-4: Definite integrals allow us to solve problems involving the accumulation of change over an interval.
CHA-4.E—Determine net change using definite integrals in applied contexts
- CHA-4.E Determine net change using definite integrals in applied contexts.
- CHA-4.E.1 The definite integral can be used to express information about accumulation and net change in many applied contexts.
- Enduring understanding CHA-4: Definite integrals allow us to solve problems involving the accumulation of change over an interval.
Topic 8.4
8.4 Finding the Area Between Curves Expressed as Functions of x
Objectives in this topic
CHA-5.A—Calculate areas in the plane using the definite integral
- CHA-5.A Calculate areas in the plane using the definite integral.
- CHA-5.A.1 Areas of regions in the plane can be calculated with definite integrals.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.5
8.5 Finding the Area Between Curves Expressed as Functions of y
Objectives in this topic
CHA-5.A—Calculate areas in the plane using the definite integral—Topic 8.5
- CHA-5.A Calculate areas in the plane using the definite integral.
- CHA-5.A.2 Areas of regions in the plane can be calculated using functions of either x or y.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.6
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
Objectives in this topic
CHA-5.A—Calculate areas in the plane using the definite integral—Topic 8.6
- CHA-5.A Calculate areas in the plane using the definite integral.
- CHA-5.A.3 Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.7
8.7 Volumes with Cross Sections: Squares and Rectangles
Objectives in this topic
CHA-5.B—Calculate volumes of solids with known cross sections using definite integrals
- CHA-5.B Calculate volumes of solids with known cross sections using definite integrals.
- CHA-5.B.1 Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.8
8.8 Volumes with Cross Sections: Triangles and Semicircles
Objectives in this topic
CHA-5.B—Calculate volumes of solids with known cross sections using definite integrals—Topic 8.8
- CHA-5.B Calculate volumes of solids with known cross sections using definite integrals.
- CHA-5.B.2 Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes.
- CHA-5.B.3 Volumes of solids with semicircular and other geometrically defined cross sections can be found using definite integrals and the area formulas for these shapes.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.9
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
Objectives in this topic
CHA-5.C—Calculate volumes of solids of revolution using definite integrals
- CHA-5.C Calculate volumes of solids of revolution using definite integrals.
- CHA-5.C.1 Volumes of solids of revolution around the x- or y-axis may be found by using definite integrals with the disc method.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.10
8.10 Volume with Disc Method: Revolving Around Other Axes
Objectives in this topic
CHA-5.C—Calculate volumes of solids of revolution using definite integrals—Topic 8.10
- CHA-5.C Calculate volumes of solids of revolution using definite integrals.
- CHA-5.C.2 Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.11
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
Objectives in this topic
CHA-5.C—Calculate volumes of solids of revolution using definite integrals—Topic 8.11
- CHA-5.C Calculate volumes of solids of revolution using definite integrals.
- CHA-5.C.3 Volumes of solids of revolution around the x- or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.12
8.12 Volume with Washer Method: Revolving Around Other Axes
Objectives in this topic
CHA-5.C—Calculate volumes of solids of revolution using definite integrals—Topic 8.12
- CHA-5.C Calculate volumes of solids of revolution using definite integrals.
- CHA-5.C.4 Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.
- Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
Topic 8.13
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
Objectives in this topic
CHA-6.A—Determine the length of a curve in the plane defined by a function, using a definite integral
- CHA-6.A Determine the length of a curve in the plane defined by a function, using a definite integral.
- CHA-6.A.1 The length of a planar curve defined by a function can be calculated using a definite integral.
- Enduring understanding CHA-6: Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.