internet audience: "thank you so much, professor leonard!! you literally saved my grade in engineering math 😭" his students, specifically that _one dude_ : "yeah, i did your way, but i got no results"
The Integral 0:30 > The Area Formula restated 1:10 Definite Integrals 4:50 > geometrical interpretation 7:00 > Properties 16:00 >w/ Example 27:00 Note: Riemann Sums are the addition of n number of rectangles to approximate the area. Taking the limit of the sum adds an infinite number of rectangles, which gives the area. This is equivalent to taking the integral of the function. A greater n gives a closer approximation of the true area.
SERIOUSLY THANK YOU. I was SO LOST IN CLASS and FELT LIKE MAYBE MATH ISNT for Me or Im NOT SMART ENOUGH but you taught me so well. Time to ACE my EXAM. Also you teach very well(passionately) and enjoyed your class i wish more teachers taught with love. Definitely and Infinitely appreciate it.
Hey Prof Leonard, I just wanted to that your calc 2 video series will be the only reason I will pass this course. Your videos are still helping students succeed even 9 years after the videos were posted!
I procrastinate to say this but I m a jee aspirant, wants to become biotechnologist and my tution don't make calculas more interesting than him so I decided to walkout tutions in order to learn from him. You will be remembered sir for this!🙏
I love your videos, they help me do well in quizzes and exams. I will support your channel when I have the means to do so one day, keep doing what you do :;) Love from Pakistan
Area of triangle @ 10:50 is wrong, it should be (3*3)/2 and not (3*2)/2. Class is really not paying attention :P EDIT: ofc he fixes it seconds later...
So..., since the negative of an area implies the area under it (or a reflexion across the x axis) does that mean that the area implies a position on the graph, rather than being indicative of the total space inside an interval? (I'm not too knowledgable on what areas are either, so bear with me)
I think you can think of 'n' as the number of rectangles you would be making under a curve. So basically, you're making the number of rectangles go to infinity. Negative infinity wouldn't really make sense in that context, although I think someone else can provide a more rigorous answer to your question.
try watching some of his other videos for scope/backround, if you have already done that and still do not get it then review your notes. If you still do not understand try practice problems or review your precalc. By now you should understand, if not then all hope is lost
I think you're confusing angles with area. We're not concerned with angles here, we're concerned with finding the area. The equation sqrt (1-x^2) basically gives us a semi-circle and we want to find the area of half of the semi-circle (a quarter of a circle). Since we know the area of a circle is pi*radius^2, to get the area of a quarter of a circle, all we need to do is divide by four, hence, the answer is pi/4.
