Integral Introduction via Work

Work done by a non-constant force is introduced using the definite integral, replacing the dot product equation which only works for constant forces. The integral is defined as the anti-derivative and as the area "under" a function, built by summing an infinite number of infinitesimally small force-versus-position rectangles, and several basic power rule integration examples lead into definite integral area calculations for increasingly complex functions. The lesson returns to its starting point with a variable force example, showing why area above the x-axis represents positive work and area below the x-axis represents negative work. Want Lecture Notes? http://www.flippingphysics.com/integr... Next Video: Hooke's Law Introduction - Force of a Spring https://www.flippingphysics.com/hooke... Previous Video: Work as the Dot Product https://www.flippingphysics.com/work-... Thank you to Mrs. Zeiler and the rest of my wonderful Patreon supporters. Please consider supporting me monthly at   / flippingphysics   Thank you to Jorge Monteiro, Julie Langenbrunner, John Paul Nichols, and Scott Carter for being my Quality Control Team for this video. http://flippingphysics.com/quality-co... Content Times: 0:00 Work and Integral Introduction 1:48 Integral Definition 4:23 Integral Math Examples 7:34 Area Example #1 9:39 Area Example #2 10:48 Area Example #3 12:36 Work Example 17:08 Defining Area "Under" a Function #Work #Integral #Calculus Curriculum Standards: AP Physics C: Mechanics, Topic 3.2 Work 3.2.A.1: "Work is the amount of energy transferred into or out of a system by a force exerted on that system over a distance." 3.2.A.2: "Work is a scalar quantity that may be positive, negative, or zero." 3.2.A.3: "The work done on an object by a variable force is calculated using W = ∫ F(r)·dr from point a to point b, where the integral is taken over the path from a to b." AP Math (Precalculus/Calculus), Unit 6 Integration and Accumulation of Change, Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals FUN-6.B.3: "If f is continuous on the interval [a, b] and F is an antiderivative of f, then the integral of f(x) dx from a to b equals F(b) - F(a)." IB Mathematics, Topic SL 5 Calculus SL 5.10: "Indefinite integral of x^n (n ∈ Q), sin x, cos x, 1/x, e^x, and their sum and difference. The composites of any of these with the linear function ax + b." SL 5.11: "Definite integrals, including analytical approach and approximation using technology. Areas under a curve (between the curve and the x-axis). Areas between curves." JEE Main Mathematics, Unit 8 Integral Calculus Unit 8: "Integral as an anti-derivative...Integration by substitution, by parts and by partial fractions...The fundamental theorem of calculus, properties of definite integrals." Australian Curriculum Mathematics, Topic Integrals ACMMM125: "interpret the definite integral ∫[a,b] f(x)dx as area under the curve y = f(x) when f(x) is positive." ACMMM131: "understand the formula ∫[a,b] f(x)dx = F(b) - F(a) and use it to calculate definite integrals." ACMMM133: "calculate total change by integrating instantaneous or marginal rate of change."