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Introduction to integral calculusDefinite integrals introExploring accumulation of changeWorked example: accumulation of change
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Riemann approximation introductionOver- and under-estimation of Riemann sumsLeft & right Riemann sumsWorked example: finding a Riemann sum using a tableWorked example: over- and under-estimation of Riemann sumsMidpoint sumsTrapezoidal sumsUnderstanding the trapezoidal ruleRiemann sums reviewMotion problem with Riemann sum approximation
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Left & right Riemann sumsGet 3 of 4 questions to level up!
Midpoint & trapezoidal sumsGet 3 of 4 questions to level up!
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Summation notationSummation notationWorked examples: Summation notation
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Riemann sums in summation notationRiemann sums in summation notationWorked example: Riemann sums in summation notationMidpoint and trapezoidal sums in summation notationRiemann sums in summation notation: challenge problem
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Definite integral as the limit of a Riemann sumDefinite integral as the limit of a Riemann sumWorked example: Rewriting definite integral as limit of Riemann sumWorked example: Rewriting limit of Riemann sum as definite integral
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The fundamental theorem of calculus and accumulation functionsFunctions defined by definite integrals (accumulation functions)Finding derivative with fundamental theorem of calculusFinding derivative with fundamental theorem of calculus: chain rule
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Interpreting the behavior of accumulation functionsInterpreting the behavior of accumulation functions
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Negative definite integralsFinding definite integrals using area formulasDefinite integral over a single pointIntegrating scaled version of functionSwitching bounds of definite integralIntegrating sums of functionsWorked examples: Finding definite integrals using algebraic propertiesDefinite integrals on adjacent intervalsWorked example: Breaking up the integral's intervalWorked example: Merging definite integrals over adjacent intervalsFunctions defined by integrals: switched intervalFinding derivative with fundamental theorem of calculus: x is on lower boundFinding derivative with fundamental theorem of calculus: x is on both boundsFunctions defined by integrals: challenge problemDefinite integrals properties review
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The fundamental theorem of calculus and definite integralsAntiderivatives and indefinite integralsProof of fundamental theorem of calculus
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Reverse power ruleIndefinite integrals: sums & multiplesRewriting before integratingRewriting before integrating: challenge problemReverse power rule review
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Reverse power ruleGet 3 of 4 questions to level up!
Reverse power rule: sums & multiplesGet 3 of 4 questions to level up!
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Indefinite integral of 1/xIndefinite integrals of sin(x), cos(x), and eˣCommon integrals review
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Indefinite integrals: eˣ & 1/xGet 3 of 4 questions to level up!
Indefinite integrals: sin & cosGet 3 of 4 questions to level up!
Integrating trig functionsGet 5 of 7 questions to level up!
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Definite integrals: reverse power ruleDefinite integral of rational functionDefinite integral of radical functionDefinite integral of trig functionDefinite integral involving natural logDefinite integral of piecewise functionDefinite integral of absolute value function
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Definite integrals: reverse power ruleGet 3 of 4 questions to level up!
Definite integrals: common functionsGet 3 of 4 questions to level up!
Definite integrals of piecewise functionsGet 3 of 4 questions to level up!
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𝘶-substitution intro𝘶-substitution: multiplying by a constant𝘶-substitution: defining 𝘶𝘶-substitution: defining 𝘶 (more examples)𝘶-substitution𝘶-substitution: rational function𝘶-substitution: logarithmic function𝘶-substitution warmup𝘶-substitution: definite integrals𝘶-substitution with definite integrals𝘶-substitution: definite integral of exponential function𝘶-substitution: special application𝘶-substitution: double substitution𝘶-substitution: challenging application
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𝘶-substitution: defining 𝘶Get 3 of 4 questions to level up!
𝘶-substitution: indefinite integralsGet 3 of 4 questions to level up!
𝘶-substitution: definite integralsGet 3 of 4 questions to level up!
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Integration using long divisionIntegration using completing the square and the derivative of arctan(x)
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Integration using long divisionGet 3 of 4 questions to level up!
Integration using completing the squareGet 3 of 4 questions to level up!
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Integral of cos^3(x)Integral of sin^2(x) cos^3(x)Integral of sin^4(x)
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Introduction to trigonometric substitutionSubstitution with x=sin(theta)More trig sub practiceTrig and u substitution together (part 1)Trig and u substitution together (part 2)Trig substitution with tangentMore trig substitution with tangentLong trig sub problem
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Trigonometric substitutionGet 3 of 4 questions to level up!
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Integration by parts introIntegration by parts: ∫x⋅cos(x)dxIntegration by parts: ∫ln(x)dxIntegration by parts: ∫x²⋅𝑒ˣdxIntegration by parts: ∫𝑒ˣ⋅cos(x)dxIntegration by parts: definite integralsIntegration by parts challengeIntegration by parts review
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Integration by partsGet 3 of 4 questions to level up!
Integration by parts: definite integralsGet 3 of 4 questions to level up!
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Integration with partial fractions
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Integration with partial fractionsGet 3 of 4 questions to level up!
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Introduction to improper integralsDivergent improper integralImproper integrals reviewImproper integral with two infinite bounds
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Improper integralsGet 3 of 4 questions to level up!
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Proof of fundamental theorem of calculusIntuition for second part of fundamental theorem of calculus

About this unit

The definite integral of a function gives us the area under the curve of that function. Another common interpretation is that the integral of a rate function describes the accumulation of the quantity whose rate is given. We can approximate integrals using Riemann sums, and we define definite integrals using limits of Riemann sums. The fundamental theorem of calculus ties integrals and derivatives together and can be used to evaluate various definite integrals.