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Integral Calculus: Math: Uncharted

Courses MathIntegral Calculus

Integral Calculus

Adding up the infinitely small: areas that become exact, the theorem that ties them to derivatives, and the techniques that make integrals yield.

Math
Unit 7 of 7
40 lessons
346 min
860 XP

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Lessons

40 total

1Area You Cannot CountRecognize rectangles as a repeatable way to estimate area under a curved boundary.InteractiveConcept8 min20 XPOpen
2Rectangles Under the CurveSet midpoint rectangle heights from a plotted function.InteractiveConcept8 min20 XPOpen
3More Rectangles, Less ErrorExplain why a finer partition tracks a curved region more closely.InteractiveConcept8 min20 XPOpen
4Left, Right, MiddleCompare left, right, and midpoint sampling on an increasing curve.InteractiveConcept7 min20 XPOpen
5Riemann WorkoutBuild and total rectangle sums on linear and curved graphs.InteractivePractice10 min20 XPOpen
6Signed AreaCombine regions above and below the axis as a net accumulated total.InteractiveConcept8 min20 XPOpen
7Accumulate the FlowUse a running total to identify when accumulated flow passes a threshold.InteractiveConcept8 min20 XPOpen
8The Integral SymbolInterpret the bounds, integrand, differential, and integral sign in definite notation.InteractiveConcept7 min20 XPOpen
9Accumulation Mastery CheckCoordinate midpoint heights with the numerical total of their rectangle areas.InteractiveMastery check10 min40 XPOpen
10Undo the DerivativeVerify a power antiderivative by differentiating its candidate.InteractiveConcept8 min20 XPOpen
11The Constant Nobody SawUse an initial value to pin the constant in an antiderivative.InteractiveConcept8 min20 XPOpen
12Antiderivative WorkoutRecover pinned antiderivatives of powers, trigonometric functions, and exponentials.InteractivePractice10 min20 XPOpen
13Initial ValueRecover a position function from velocity and a starting position.InteractiveConcept8 min20 XPOpen
14Check by DifferentiatingVerify a chain-rule antiderivative by simplifying its derivative.InteractivePractice8 min20 XPOpen
15Two Candidates, One IntegrandIdentify all candidates whose derivatives equal one integrand.InteractiveConcept7 min20 XPOpen
16Rates BackwardAccumulate a changing production rate from a known starting total.InteractiveConcept8 min20 XPOpen
17Two Problems, One AnswerConnect a shaded area with an antiderivative difference.InteractiveConcept9 min20 XPOpen
18The Fundamental TheoremEvaluate a definite integral by subtracting endpoint values of an antiderivative.InteractiveConcept8 min20 XPOpen
19Evaluate the IntegralEvaluate polynomial definite integrals on integer intervals.InteractivePractice10 min20 XPOpen
20Area, ExactlyCompare a rectangle estimate with an exact Fundamental Theorem value.InteractivePractice9 min20 XPOpen
21Fundamental Theorem Mastery CheckUse an antiderivative to turn a definite accumulation into one exact number.InteractiveMastery check10 min40 XPOpen
22Substitute and SeeVerify a substitution-pattern antiderivative through the chain rule.InteractiveConcept8 min20 XPOpen
23Pick the Inside FunctionChoose an inside function whose differential exposes a substitution.InteractiveConcept7 min20 XPOpen
24Substitution WorkoutFind pinned antiderivatives with one clean substitution.InteractivePractice10 min20 XPOpen
25Trig IntegralsVerify a trigonometric antiderivative with a chain-rule simplification.InteractiveConcept8 min20 XPOpen
26Exponential IntegralsIntegrate an exponential with a linear exponent and pin its constant.InteractiveConcept8 min20 XPOpen
27Split It UpRewrite an integrand into simpler pieces before integrating.InteractiveConcept8 min20 XPOpen
28Techniques WorkoutMix direct antiderivatives, substitutions, and differentiation checks.InteractivePractice10 min20 XPOpen
29Which TechniqueDistinguish direct antiderivatives from one-step substitutions.InteractivePractice8 min20 XPOpen
30Area Between CurvesCalculate the area of a strip by subtracting lower accumulation from upper accumulation.InteractiveConcept9 min20 XPOpen
31Between WorkoutUse definite-integral differences to measure regions between graphs.InteractivePractice10 min20 XPOpen
32The Average HeightSet a constant height whose rectangle has the same area as a curved region.InteractiveConcept9 min20 XPOpen
33Distance from SpeedRead distance as accumulated area under a speed-time curve.InteractiveConcept8 min20 XPOpen
34Volume by SlicesEvaluate a given disk-method integral for a solid of revolution.InteractiveConcept9 min20 XPOpen
35Volume WorkoutCompute volumes from supplied disk and washer setups.InteractivePractice10 min20 XPOpen
36Work to Fill a TankEvaluate a physical work integral with a changing lift distance.InteractiveConcept9 min20 XPOpen
37A First Differential EquationSelect the exponential function fixed by a rate law and initial value.InteractiveConcept9 min20 XPOpen
38Growth from a RateLocate a population peak from the zero crossing of its growth rate.InteractiveConcept8 min20 XPOpen
39Applied Integrals WorkoutMove fluently among area, average value, distance, volume, and work totals.InteractivePractice10 min20 XPOpen
40Integral Calculus Mastery CheckVerify a composite antiderivative and connect it with a definite accumulated value.InteractiveMastery check11 min40 XPOpen

Differential Calculus