# Applications of integration

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑5 (EU), CHA‑6 (EU)

## Finding the average value of a function on an interval

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑4.B (LO), CHA‑4.B.1 (EK)
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Average value over a closed intervalCalculating average value of function over intervalMean value theorem for integrals
Practice
Average value of a functionGet 3 of 4 questions to level up!

## Connecting position, velocity, and acceleration functions using integrals

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑4.C (LO), CHA‑4.C.1 (EK)
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Motion problems with integrals: displacement vs. distanceAnalyzing motion problems: positionAnalyzing motion problems: total distance traveledMotion problems (with definite integrals)Worked example: motion problems (with definite integrals)Average acceleration over interval
Practice
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

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑4.D (LO), CHA‑4.D.1 (EK), CHA‑4.D.2 (EK), CHA‑4.E (LO), CHA‑4.E.1 (EK)
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Area under rate function gives the net changeInterpreting definite integral as net changeWorked examples: interpreting definite integrals in contextAnalyzing problems involving definite integralsAnalyzing problems involving definite integralsWorked example: problem involving definite integral (algebraic)
Practice
Interpreting definite integrals in contextGet 3 of 4 questions to level up!
Analyzing problems involving definite integralsGet 3 of 4 questions to level up!
Problems involving definite integrals (algebraic)Get 3 of 4 questions to level up!

## Finding the area between curves expressed as functions of x

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.A (LO), CHA‑5.A.1 (EK)
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Area between a curve and the x-axisArea between a curve and the x-axis: negative areaArea between curvesWorked example: area between curvesComposite area between curves
Practice
Area between a curve and the x-axisGet 3 of 4 questions to level up!
Area between two curves given end pointsGet 3 of 4 questions to level up!
Area between two curvesGet 3 of 4 questions to level up!

## Finding the area between curves expressed as functions of y

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.A (LO), CHA‑5.A.2 (EK)
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Area between a curve and the 𝘺-axisHorizontal area between curves
Practice
Horizontal areas between curvesGet 3 of 4 questions to level up!

## Finding the area between curves that intersect at more than two points

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.A (LO), CHA‑5.A.3 (EK)
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## Volumes with cross sections: squares and rectangles

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.B (LO), CHA‑5.B.1 (EK)
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Volume with cross sections: introVolume with cross sections: squares and rectangles (no graph)Volume with cross sections perpendicular to y-axis
Practice

## Volumes with cross sections: triangles and semicircles

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.B (LO), CHA‑5.B.2 (EK), CHA‑5.B.3 (EK)
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Volume with cross sections: semicircleVolume with cross sections: triangle
Practice

## Volume with disc method: revolving around x- or y-axis

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.C (LO), CHA‑5.C.1 (EK)
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Disc method around x-axisGeneralizing disc method around x-axisDisc method around y-axis
Practice
Disc method: revolving around x- or y-axisGet 3 of 4 questions to level up!

## Volume with disc method: revolving around other axes

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.C (LO), CHA‑5.C.2 (EK)
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Disc method rotation around horizontal lineDisc method rotating around vertical lineCalculating integral disc around vertical line
Practice
Disc method: revolving around other axesGet 3 of 4 questions to level up!

## Volume with washer method: revolving around x- or y-axis

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.C (LO), CHA‑5.C.3 (EK)
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Solid of revolution between two functions (leading up to the washer method)Generalizing the washer method
Practice
Washer method: revolving around x- or y-axisGet 3 of 4 questions to level up!

## Volume with washer method: revolving around other axes

AP Calc: CHA (BI), CHA‑5 (EU), CHA‑5.C (LO), CHA‑5.C.4 (EK)
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Washer method rotating around horizontal line (not x-axis), part 1Washer method rotating around horizontal line (not x-axis), part 2Washer method rotating around vertical line (not y-axis), part 1Washer method rotating around vertical line (not y-axis), part 2
Practice
Washer method: revolving around other axesGet 3 of 4 questions to level up!

## The arc length of a smooth, planar curve and distance traveled

AP Calc: CHA (BI), CHA‑6 (EU), CHA‑6.A (LO), CHA‑6.A.1 (EK)
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Arc length introWorked example: arc length
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Arc lengthGet 3 of 4 questions to level up!

## Calculator-active practice

AP Calc: CHA (BI), CHA‑4 (EU), CHA‑4.B (LO), CHA‑4.C (LO), CHA‑4.E (LO)
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