2nd Fundamental Theorem of Calculus & Mean Value Theorem Made Easy - Step by Step

#Calculus #FundamentalTheoremOfCalculus #SecondFundamentalTheorem #MeanValueTheoremForIntegrals #AverageValueOfAFunction #DefiniteIntegrals #Integration #Calculus1 #IntegralCalculus #MathTutorial #LearnCalculus #MathAndEngineeringMadeEasy Lesson 19 of the Calculus 1 series: two short, high-value topics. The derivative of an integral (the Second Fundamental Theorem of Calculus), and the average value of a function (the Mean Value Theorem for Integrals). In this session: The accumulation function: why ∫ₐˣ f(t) dt "accumulates" area as x increases The theorem: d/dx ∫ₐˣ f(t) dt = f(x), so the derivative and the integral cancel Why the integration variable is t (or any letter other than x) The chain-rule version when the upper limit is a function: d/dx ∫ₐ^g(x) f(t) dt = f(g(x)) · g′(x) Example 1: d/dx ∫₁ˣ (1 − t³) dt Example 2: a variable upper limit of x², with the chain rule The average value of a function: f_avg = 1/(b − a) · ∫ₐᵇ f(x) dx, explained as "area under the curve = area of a rectangle with the average height" Example 3: the average value of sin x on [0, π] (answer: 2/π) Example 4: the average value of √x on [0, 16] (answer: 8/3), checked on a graphing calculator This builds on Lesson 18 (definite integrals and the first Fundamental Theorem), so watch that first if you haven't yet. Next up: integrals involving inverse trig functions. These results help connect integration and differentiation while providing an efficient method for finding the average value of a function. Subscribe to Math and Engineering Made Easy for more accessible, step-by-step tutorials in calculus, mathematics, and engineering. Have a question? Leave a comment below! #Calculus #FundamentalTheoremOfCalculus #SecondFundamentalTheorem #MeanValueTheoremForIntegrals #AverageValueOfAFunction #DefiniteIntegrals #Integration #Calculus1 #IntegralCalculus #MathTutorial #LearnCalculus #MathAndEngineeringMadeEasy