Calculus 1 Lecture 5.1: Finding Area Between Two Curves

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Calculus for Engineers II class. Struggle-cuddling these videos rn while I try not to go rock back and forth in the corner of my room lmao

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This is such a strange lie... it's not possible to fail the first exam and end the class with a 97%. You'd have had to literally get 100% on everything else and the first test would have still had to have been only like 10% of your final grade.

[@GodsNode]

0:00 Setup 14:40 Ex 1 21:40 Ex 2 38:50 Ex 3 42:00 Ex 4 51:50 Ex 5 1:00:19 Ex 6 (in terms of y AND x) - 1:07:45 What are you integrating w/respect to? The hard way. 1:27:23 The easy way

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[@taekwondotime]

*TIP:* In Calculus 3, these concepts will be extended to find the area between 3-D surfaces and 3-D objects.

[@taekwondotime]

TIP: @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)

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Dear Brofessor Leonard, I'm on summer break between Calc 1 and Calc 2 and I've been going through your lectures of both. It's wonderful that you never assume that your students know things. You've cleared up so many random areas of confusion about logarithms, exponent rules, trig/inverse-trig, properties of this or that thing, all while watching videos that weren't even covering that topic.

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[@taekwondotime]

*TIP:* @36:00 It helps to separate the constants from the variables when you go to evaluate the integral. For the first term in the integral, begin by writing 1/2 x^2 instead of x^2/2. That way you can pull out the 1/2 and be left with evaluating 3^2 - (-2)^2 which is much easier. You get 9 - 4 which equals 5 and then you multiply by 1/2 to get 5/2. Every time you see a constant, get rid of it before plugging in your bounds to evaluate and you'll make the math a whole lot easier and save yourself from doing the same work over and over again. :)

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my final is tomorrow and this helped me understand area between two curves so much better!

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[@dawnfantasy]

0:46:00, IMO, if x^3-x is an odd function, the net signed area is 0, and total area is 1/2.

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[@Neme112]

At 58:28 he says if the cars pass each other (or I guess if the velocities pass each other since that's what's in the graph), you'd have to make separate integrals for each interval. That's *NOT* true. That would be the case if we wanted to know just the abstract concept of the total area between those curves, but in order to get the final distance between the cars, net signed area is actually what we want. In the real world, it is true that if one car is faster than the other and then vice versa, the distance between them "cancels out", just like net signed area. If we actually did separate integrals and got the total area, we'd get the wrong answer. That wouldn't be the distance between the cars.

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book name plz

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1:12:35 Why doesn't he use y=(x-2) in the first equation x=(x-2)^2 it gives the same value of co ordinates (4,2) (1,-1)

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if i am not wrong, he was explaining the Rolle’s Theorem when he said when f is continuous over the closed intervals [a,b] and differentiate over open intervals (a,b), then there must exist a number c when f’(k)=0.

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[@rualecreatures6]

My Calc 2 teacher at the university of Akron is probably the worst math teacher I have ever had I sat through 20 minutes of this lecture and understood more from this video than 2 hours in my actual lecture.

[@ishx_03]

1:19:41 you did not integrate lower second tringle . but when i do integration by taking lower second triangle answer is 9÷ 2.

[@leciannaicker7256]

🎯 Key points for quick navigation: 00:00 *📐 Calculus 1 Lecture 5.1: Introduction to Finding Area Between Two Curves* - Introduces the concept of finding the area between two curves using integrals. - Explains that the area between two curves is determined by subtracting the integral of the lower curve from the integral of the upper curve over a defined interval. - Emphasizes that the resulting area is always positive if the upper curve is consistently above or equal to the lower curve over the interval. 03:09 *🧮 Understanding the Requirements for Finding Area Between Two Curves* - Requires the upper curve to be greater than or equal to the lower curve throughout the interval for a positive area. - Illustrates with graphical examples where the condition is met versus where it isn't. - Emphasizes the necessity of consistent positioning (upper or equal) of the curves for correct area calculations. 06:09 *📝 Application of Integrals for Finding Area Between Two Curves* - Demonstrates the application of definite integrals to find the area between two curves. - Shows how to combine integrals when both curves have the same bounds of integration. - Explains the process of setting up integrals for upper and lower functions and simplifying calculations accordingly. 25:13 *📈 Understanding quadratic equations and factoring* - Explains how to factor quadratic equations such as \( x^2 - x - 6 = 0 \). - Demonstrates the process of factoring and verifying solutions algebraically. - Highlights the importance of identifying solutions correctly in algebraic equations. 26:02 *📊 Determining bounds for integration based on intersections* - Discusses how to find the bounds of integration using intersections of graphs. - Emphasizes that intersections define the limits of integration for calculating areas. - Illustrates with examples how multiple intersections affect the number of integral calculations needed. 27:56 *🔍 Identifying which function is on top for integration* - Teaches methods to determine which function is on top within a given interval. - Emphasizes the role of function evaluations to establish dominance in integration setups. - Shows how choosing a test point within the interval helps in identifying the superior function for integration. 30:33 *📐 Setting up integrals to find area between curves* - Guides through the process of setting up integrals to calculate the area between two curves. - Demonstrates the integration setup based on which function is on top for each interval. - Highlights the importance of correctly setting up integrals using determined intervals and function dominance. 52:40 *🏎️ Velocity comparison and area between curves* - Explains the concept of comparing velocities over time and finding the area between two velocity curves. - Area between curves in this context represents the distance one car (me) is ahead of another (you). - Velocity curves allow for precise measurement of who wins a race and by how much, unlike simple velocity comparisons. 54:06 *📏 Calculating distance using area between curves* - Demonstrates how the area between two velocity curves corresponds to the cumulative distance covered by each car. - Integrating velocity curves yields distance traveled, providing a clear measure of how far one car is ahead of another. - This method offers a detailed comparison of positional differences over time, enhancing race analysis beyond simple speed comparisons. 01:00:24 *🔍 Integrating curves for area calculations* - Discusses the process of integrating curves with respect to the x-axis to find areas bounded by multiple functions. - Emphasizes the need to express functions in terms of x to correctly set up integrals for area calculations. - Points out the importance of identifying intersection points and understanding function behavior across different intervals for accurate integration. 01:17:40 *📐 Calculating area between curves* - Explanation of integrating to find the area between curves in calculus, - Importance of correctly identifying upper and lower functions for the integral, - Steps to set up and solve the integral for area calculation. 01:19:25 *📊 Simplification before integration* - Importance of simplifying expressions before integrating, - Tips on combining like terms and reducing complexity, - Demonstrating step-by-step integration and simplification. 01:27:18 *🔄 Changing variables for easier integration* - Explanation of changing variables to simplify integrals, - Demonstrating how to switch from x-axis to y-axis calculations, - Applying variable changes to specific calculus problems for clarity. Made with HARPA AI

[@marxho]

Awesome lecture!!!

[@uiticus]

At 6:30 the area between 2 curves is the difference between the 2 integrals of the respective functions. Why would not the C get involved in this case, since the C value is different for these 2 functions?

[@uiticus]

Yes, I know. Thank you.

[@kalebscott282]

When instead of teaching, your professor just links this guy's video.