Wemazi

AP Calculus · Unit 8

Applications of Integrals

Average value, motion, area and volume.

Topic 8.1Average value of a function

  1. Set up the average-value calculationDivide the definite integral of the function by the interval length.
  2. Calculate an average valueIntegrate or use supplied area or tabular information and keep the original quantity's units.
  3. Distinguish average value from average rateSelect the correct expression for what the question asks.
  4. Find where the function equals its average valueSolve for matching function values and identify a guarantee only when its conditions hold.

Topic 8.2Recover motion by integration

  1. Recover velocity or positionIntegrate the given rate and use the corresponding initial value.
  2. Calculate displacementIntegrate signed velocity over the interval.
  3. Calculate total distanceFind velocity sign changes and integrate speed or sum absolute displacements.
  4. Distinguish final position, displacement, and distanceAnswer the requested quantity rather than substituting another motion total.

Topic 8.3Accumulation in applications

  1. Choose a rate and accumulation intervalTranslate the context into an appropriate definite integral with units.
  2. Combine inflow and outflowIntegrate the net rate with the correct signs.
  3. Recover an amount from a starting valueAdd the accumulated change and respect relevant physical constraints.
  4. Find when an accumulated amount is greatest or leastCheck net-rate sign changes, endpoints, and candidate amount values.
  5. Interpret the result in contextDistinguish amount, net change, and total flow and select a justified statement.

Topic 8.4Area using vertical slices

  1. Find the horizontal limits of the regionSolve intersections or read the given left and right boundaries.
  2. Identify the upper and lower curvesDetermine the vertical slice height over the interval.
  3. Set up an integral with respect to xUse upper minus lower with the correct bounds.
  4. Calculate geometric areaEvaluate the integral and report a nonnegative result in square units.

Topic 8.5Area using horizontal slices

  1. Express the boundaries in terms of yChoose valid branches when rewriting curves for horizontal slices.
  2. Find the vertical limits of the regionDetermine the lower and upper y-values from boundaries or intersections.
  3. Set up an integral with respect to yUse the right x-value minus the left x-value, with the correct bounds.
  4. Calculate area using horizontal slicesEvaluate the setup and distinguish x-width from y-height.

Topic 8.6Area with repeated crossings

  1. Find every relevant crossingLocate all intersection inputs inside the region's bounds.
  2. Determine which curve is on top in each intervalTrack order changes instead of assuming one order throughout.
  3. Split the area calculation correctlyUse separate integrals or an equivalent valid absolute-value setup.
  4. Combine the region areasAdd nonnegative pieces rather than allowing signed cancellation.

Topic 8.7Square and rectangular cross sections

  1. Find the base width of a cross sectionRead the distance between region boundaries in the stated slicing direction.
  2. Build the square or rectangle areaSquare the side for a square, or multiply width by the specified rectangle height.
  3. Set up the volume integralIntegrate cross-sectional area over the correct input interval.
  4. Calculate and interpret volumeEvaluate the integral and report cubic units.

Topic 8.8Triangular and semicircular cross sections

  1. Identify the dimensions of the sliceDistinguish diameter from radius and base from height or hypotenuse.
  2. Build a triangular cross-sectional areaUse the specified triangle geometry to express area from the base-region width.
  3. Build a semicircular cross-sectional areaConvert the diameter to a radius before applying the half-circle area formula.
  4. Integrate the slice areasUse the correct direction, bounds, and cubic units for the volume.

Topic 8.9Disks around a coordinate axis

  1. Choose slices perpendicular to the axisMatch a horizontal or vertical rotation axis with the correct integration variable.
  2. Measure the disk radiusUse the distance from the coordinate axis to the boundary.
  3. Build the disk areaSquare the radius and multiply by pi, with no inner hole.
  4. Set up and evaluate the volumeUse the region's correct bounds and report cubic units.

Topic 8.10Disks around a shifted axis

  1. Locate the shifted rotation axisIdentify the horizontal or vertical line about which the region turns.
  2. Measure the radius from that lineUse a nonnegative distance rather than the raw function height or width.
  3. Set up the disk integralMatch the variable, squared radius, and region bounds.
  4. Calculate the volumeEvaluate the integral and interpret its cubic units.

Topic 8.11Washers around a coordinate axis

  1. Identify the outer and inner radiiMeasure both distances from the coordinate axis and determine their order.
  2. Build a washer areaSubtract the squared inner radius from the squared outer radius. Then multiply by pi.
  3. Choose bounds and slicing directionUse slices perpendicular to the rotation axis over the correct region.
  4. Set up and evaluate the volumeIntegrate the washer area rather than squaring the difference of the radii.

Topic 8.12Washers around a shifted axis

  1. Measure both radii from a shifted axisExpress distances to the outer and inner boundaries relative to the stated line.
  2. Determine radius order across the regionCheck whether the outer boundary changes or the region crosses the axis.
  3. Build the correct slice-area formulaUse a washer or disk as appropriate, splitting intervals when necessary.
  4. Set up and evaluate the volumeUse the correct variable, bounds, and squared radii.

Topic 8.13Arc length of a function graph

BC only
  1. Build the arc-length integrandCombine one plus the squared derivative under a square root.
  2. Choose the variable and boundsMatch the setup to y as a function of x or x as a function of y.
  3. Calculate or approximate the lengthSimplify and integrate when possible, or use an appropriate numerical approximation.
  4. Distinguish arc length from area and perimeterInclude closing straight segments only when the question asks for the whole perimeter.