THE INTEGRAL, together with the DERIVATIVE, is one of the most important concepts in CALCULUS. But, what is the integral? What does it mean? Why is it related to the area under the graph of a function? Is there any relationship between DERIVATIVES and INTEGRALS? In this video we will talk about the motivations that lead us to the study of the calculation of areas under the graphs of functions, which leads us to solve this problem through a fantastic idea: finding the area through approximations and using the limit we arrive at the definition of the DEFINED INTEGRAL. In addition, we will see an intuitive idea to understand the FIRST FUNDAMENTAL THEOREM OF CALCULUS, which establishes a DIRECT RELATIONSHIP with the DERIVATIVE, we also make a deduction of the SECOND FUNDAMENTAL THEOREM OF CALCULUS or the famous BARROW RULE, through which we can solve integrals more effectively. In addition, examples are given to briefly show its application and its importance in mathematics, science, and engineering. It may seem like a long video, but I think it is enough for a whole semester. If you are a student or interested in learning, I would recommend taking NOTES on the main parts and then REFLECTING on your own to better understand the topic. Creating these videos takes me quite a bit of time. If you want to help me continue creating content on YouTube, I invite you to support me on PATREON and ACCESS EXCLUSIVE BENEFITS: / bluedot96 #integrals #differentialcalculus #integralcalculus #derivatives #fundamentaltheoremcalculus #barrowsrule CHAPTERS: 00:00 Introduction 01:05 Distance traveled with constant speed 03:37 Distance traveled with variable speed 05:45 The area problem and the Riemann sum 12:20 Definition of the definite integral 13:58 Riemann sums by right, left or center 15:20 Example of the area under the curve of x ^ 2 by the definition of the integral 21:04 Relationship between INTEGRALS and DERIVATIVES 24:22 Antiderivatives 27:03 First fundamental theorem of calculus 28:01 Second fundamental theorem of calculus or Barrow's rule 31:40 Example of the area under the graph of x^2 using Barrow's rule 33:02 Solution to the distance problem 34:21 Goodbye ##################################################### You can support the making of more videos of this style by being a MEMBER OF THE CHANNEL, with your help I will be able to make more videos until I complete a playlist with university mathematics from the most basic to the most complex through animations! The creation of these videos takes me quite a bit of time. If you want to help me continue creating content on YouTube, I invite you to support me on PATREON and ACCESS EXCLUSIVE BENEFITS: / bluedot96 Follow me on Twitter: / bluedot96 Follow me on Facebook: / bluedot1196 Follow me on Instagram: / pauloch96 MUSIC: Kevin Bryce: / @kevinbryce VIDEOS: https://pixabay.com/es/videos/ The final video is a fragment of the video: The story of everything - Pyrander THESE VIDEOS MAY INTEREST YOU: I RECOMMEND THESE VIDEOS: WHAT IS THE DERIVATIVE?: • The DERIVATIVE changed EVERYTHING ???? WHAT is... WHAT IS THE INTEGRAL?: • What is the INTEGRAL? MEANING of... WHAT IS A FUNCTION?: • A function is NOT A MACHINE WHAT ... FUNCTIONS FROM SCRATCH: • WHAT is a FUNCTION? ▶ CARTRIDGE PRODUCT... LINEAR FUNCTION: • What is a LINEAR FUNCTION? ▶ GRAPH... QUADRATIC FUNCTION: • WHAT is a QUADRATIC FUNCTION? ▶ Com... FUNCTIONS OF TWO VARIABLES: • WHAT is a FUNCTION OF TWO VARIABLES?... THE INTEGRAL FROM SCRATCH: • This THEOREM CHANGED EVERYTHING ???? WHAT i... DIFFERENTIATION: • This IDEA OF NEWTON FOUNDED the BASIS... RULES OF DERIVATION: • The UNKNOWN side of the DERIVATIVE ????... WHAT IS CALCULUS?: • Why did CALCULUS CHANGED EVERYTHING? ... WHAT IS A LIMIT?: • A FANTASTIC INTRODUCTION to LIMITS... THE LIMIT EXPLAINED IN DETAIL: • WHAT IS THE LIMIT OF A FUNCTION? ▶OF... LIMIT PROPERTIES: • WHAT IS BEHIND THE FAMOUS PROPERTIES... WHAT IS OPTIMIZATION?: • Why SHOULD YOU LEARN OPTIMIZATION... WHERE DO SIN, COS AND TAN COME FROM?: • REALLY UNDERSTANDING WHAT SIN, COS... UNDERSTANDING SIN, COS AND TAN: • UNDERSTANDING THE SIN, COS AND ... ***************************************************************************************************** The animations in this video were made with Manim (Mathematical animation), a library for Python created by Grant Sanderson (3Blue1Brown). Personally, I find it to be a very useful tool to be able to better explain many concepts in a visual way. If you are interested, here is the link to the Manim library for Python: https://github.com/3b1b/manim Without further ado, thank you for your attention and see you in the next video