Wemazi

AP Calculus · Unit 6

Integration

Accumulation, antiderivatives and the FTC.

Topic 6.1Build a total from a rate

  1. Represent change as a definite integralChoose the rate, limits, and units for the accumulated quantity.
  2. Read signed area from a rate graphAdd positive and negative contributions over the requested interval.
  3. Recover a final amountAdd the accumulated net change to the initial amount.
  4. Distinguish net change from total accumulationIdentify when absolute values are required and what cancellation means.

Topic 6.2Approximate an integral numerically

  1. Build left and right endpoint sumsUse each subinterval's actual width and the correct endpoint height.
  2. Build a midpoint sumEvaluate or read the rate at the midpoint of each subinterval.
  3. Use the trapezoidal ruleAverage adjacent heights and multiply by each interval's width.
  4. Determine the direction of errorUse monotonicity or concavity when enough information is given.
  5. Interpret a numerical integral in contextReport an approximation with accumulated units and avoid claiming unsupported accuracy.

Topic 6.3Translate sums and integrals

  1. Find the width and sample pointsTranslate a partition into its subinterval width and left, right, or midpoint inputs.
  2. Separate height from width in a sumIdentify the function being sampled and the interval it covers.
  3. Translate a sum limit into an integralGive a consistent integrand and pair of bounds.
  4. Build a sum for a definite integralForm a correct sample-point expression and width factor for the chosen partition.

Topic 6.4Differentiate accumulation

  1. Differentiate an accumulation with a variable endpointApply the Fundamental Theorem of Calculus at the correct endpoint.
  2. Include the chain-rule factorDifferentiate a composed endpoint such as x squared.
  3. Handle two moving endpointsSubtract the lower-bound contribution and include both endpoint derivatives.
  4. Use graph or table values in the derivativeEvaluate the integrand at the transformed inputs without first finding an antiderivative.

Topic 6.5Read an accumulation function

  1. Calculate an accumulation valueCombine signed areas with the starting value or starting integration point.
  2. Find where the accumulation increases or decreasesUse the sign of its derivative, including any additional non-integral term.
  3. Find extrema of the accumulationCheck sign changes and compare candidate accumulated values when needed.
  4. Determine the accumulation's concavityRelate the integrand's increase or decrease to the accumulation's second derivative.
  5. Distinguish the graph's height from accumulated areaAvoid confusing zeros of the integrand with zeros of the accumulation.

Topic 6.6Rearrange definite integrals

  1. Reverse or collapse integration boundsChange the sign when reversing bounds and use zero when the bounds agree.
  2. Split and combine intervalsUse additivity to recover an integral over a missing segment.
  3. Apply linearity correctlySeparate sums and constant multiples without inventing product or quotient rules for integrals.
  4. Use symmetryEvaluate or relate integrals of even and odd functions over appropriate intervals.

Topic 6.7Evaluate with antiderivatives

  1. Find a suitable antiderivativeUse the appropriate basic antiderivative for the integrand.
  2. Evaluate at the bounds in the correct orderSubtract the lower-endpoint value from the upper-endpoint value.
  3. Handle a changing formulaSplit a piecewise or absolute-value integrand at the required points.
  4. Interpret the definite resultDistinguish signed accumulation from geometric area and omit an unnecessary integration constant.

Topic 6.8Find an antiderivative

  1. Find an antiderivative of a power or sumApply reverse differentiation and handle the exponent correctly.
  2. Use basic trig, exponential, and logarithmic antiderivativesRecognize the appropriate family and its domain.
  3. Include the integration constantRepresent a family of antiderivatives rather than one unexplained choice.
  4. Determine a particular antiderivativeUse a supplied function value to solve for the constant.
  5. Check an antiderivativeDifferentiate it and compare with the original integrand.

Topic 6.9Integrate by substitution

  1. Choose a useful substitutionIdentify an inner function and its corresponding differential.
  2. Rewrite the whole integral consistentlyReplace both the integrand and differential, keeping any scale factor.
  3. Transform definite-integral boundsExpress both limits in the new variable or return fully to the original variable before evaluating.
  4. Complete and check the integrationIntegrate the transformed expression and produce the requested final form or value.

Topic 6.10Rewrite before integrating

  1. Divide or cancel algebraic factorsSimplify a rational integrand while keeping relevant domain restrictions.
  2. Rewrite radicals and powersExpand or express terms as powers that can be integrated directly.
  3. Recognize a useful trig identityRewrite the integrand using an appropriate identity.
  4. Integrate the rewritten expressionCarry the algebraic rewrite through to the correct antiderivative or definite value.

Topic 6.11Integration by parts

BC only
  1. Choose the parts of the integralSelect u and dv so that differentiation and integration make the remaining problem useful.
  2. Apply integration by partsForm the boundary or product term and the remaining integral with the correct sign.
  3. Evaluate the resulting integralComplete an indefinite or definite calculation, including constants or bounds as appropriate.
  4. Use endpoint and integral dataApply the formula when functions are supplied through data instead of explicit expressions.

Topic 6.12Linear partial fractions

BC only
  1. Prepare the rational expressionFactor the denominator and perform division first if the fraction is improper.
  2. Write the partial-fraction formUse one term for each distinct linear factor in the core AP case.
  3. Determine the coefficientsClear denominators and solve the resulting identities.
  4. Integrate the decomposed expressionInclude logarithmic absolute values and handle constants or bounds correctly.

Topic 6.13Improper integrals

BC only
  1. Identify why an integral is improperDetect infinite bounds and singularities at endpoints or inside the interval.
  2. Replace the improper part with a limitSplit interior singularities into separately converging one-sided integrals.
  3. Decide convergence or divergenceEvaluate the limits rather than canceling opposing infinities.
  4. Calculate a convergent improper integralReport a finite value only when every required part converges.
  5. Recognize standard convergence thresholdsApply the appropriate power threshold at infinity or near a singular endpoint.

Topic 6.14Choose an integration technique

  1. Recognize a direct antiderivativeIdentify a familiar pattern that needs no more elaborate technique.
  2. Choose substitution when it fitsDetect an inner function and its derivative.
  3. Choose an algebraic rewriteUse expansion, division, or an identity to simplify the integration.
  4. Choose among the remaining methods in your courseIn BC, also compare integration by parts and partial fractions with simpler methods; these are not required for AB.
  5. Carry the chosen method through correctlyProduce a correct integral value or antiderivative and respect bounds and domain.