Introduction to Definite Integrals via Physics (Distance & Speed, Displacement & Velocity)

When the velocity is positive, it is the speed and the area under the graph (definite integral) is the distance traveled. Approximating rectangles can be used to estimate the area under the curve, either with a left sum, right sum, or their average (Trapezoid Rule). If the velocity becomes negative, then the definite integral gives the displacement and it is computed as an area under the graph when the velocity is positive minus an area above the graph when the velocity is negative. Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

Calculus and Physics Problems and Solutions

🔴 Calculus 1, Lesson 1.1: Linear Functions (3Blue1Brown SoME2): https://www.youtube.com/watch?v=lmCjnxzIals

🔴 Calculus 1, Lesson 1.2: Exponential Functions: https://www.youtube.com/watch?v=47ZqW19AjD4

🔴 Calculus 1, Lesson 1.3: Quadratic Functions, Transformations of Functions, and Function Composition: https://www.youtube.com/watch?v=1T58CnSME4g

🔴 Calculus 1, Lesson 1.4: Logarithmic Functions (Intro to Horizontal Compressions & Inverse Functions): https://www.youtube.com/watch?v=oS-gvIoJf60

🔴 Calculus 1, Lesson 1.5: Trigonometric Functions and Inverse Trigonometric Functions: https://www.youtube.com/watch?v=UFYjlnGUenE

🔴 AP Calculus AB Exam Review: Practice Exam Problems & Solutions (Multiple Choice, No Calculator): https://www.youtube.com/watch?v=2AG_Dt3x7q0

🔴 AP Calculus AB Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=RLgUwP44V_M

🔴 AP Calculus BC Exam Review: Practice Exam (30 Question Multiple Choice, No Calculator): https://www.youtube.com/watch?v=1Q7ALcwT97A

🔴 AP Calculus BC Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=ZWisJEDP44w

🔴 Calculus 1 Exam 1 Review Problems and Solutions (Precalculus, Limits, Continuity, Derivatives): https://www.youtube.com/watch?v=UjmvUwvd7c4

🔴 Calculus 1 Course, Lecture 1: The Big Ideas (Rates & Areas, the Infinity Principle & Circular Area): https://www.youtube.com/watch?v=2V_ehJC5QUY

🔴 Calculus 1 Course Lecture Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MkFrKtM0S0NLu-TxZa5jBIx

#calculus #integration #integral

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(0:00) Distance as area under speed curve
(6:15) Left-hand sum vs right-hand sum
(7:52) Upper vs lower estimates
(8:26) # of rectangles = n and Δt = (b – a)/n (width of each subinterval)
(10:15) Estimate the total area (definite integral)
(16:23) Trapezoid sum (average of left sum and right sum)
(19:08) Displacement as integral of discontinuous velocity (for back and forth motion, with both positive and negative velocities)
(27:43) Displacement with a continuous velocity
(30:56) Find unknown constants in a graph (based on physical motion)
(31:46) Find distance traveled for linearly decelerating car
(33:49) Which car travels further? (Sharp velocity curve with high peak vs wider velocity curve with low peak)
(35:55) Difference between upper and lower estimates
(41:20) How big should n be to guarantee distance between upper and lower estimates is less than 0.0001?

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