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Can you do integral limits?

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I'm confused on problem 53 because the solution uses division rather than subtraction

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With the idea of the reman sum, would it be possible to take a Riemann sum of two or more functions that intersect?  So that the area is between the intersections? Or is the idea of a Riemann sum and the fundamental theorem of calculus only applicable to functions that have continuity?

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We encountered and used the idea of limits before looking at the ideas of both derivatives and definite integrals. Will the concept of limits give us another mathematical construct and/or will we learn more about limits?

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One thing I'm interested about is what methods are used to interpret the definite integral of a graph without a given function? Is it just a process of manual estimation. How many different applications of the definite integral are there? I know about the application of net change and of volume but what are others?

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What would happen if instead of taking the area underneath the curve between two specific points, and stretched it out to infinity? Would we can take the limit as the function approaches infinity to calculate the definite integral, such as in f(x)= 1/x. As x approaches infinity, y approaches 0. What would happen if we took it from negative to positive infinity (what would that represent in application mathematics). I figured out that the first and third quadrants give positive area (the negative height and width create a positive area), and the second and forth give a negative area. In functions that are reflected across the y axis, would we get an area of zero since the two areas cancel out?

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This isn't super mathy per say but when I tried to graph the function from the last problem on the homework the first time it assumed xe was a variable because I used to e from my key pad and not the one on the screen which I thought was weird and good to know because I was pretty confused about why it was like that for a second.

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Can you find the definite integral of a relation that is not a a function? This link this the only thing that came up about relations that aren't function but I'm not sure how trustworthy this site it.

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I wanna take the definite integral of that relation but I guess I am not sure of the notation or if it is even possible...and from x(or/and y?) = -1 to 1

Z is just some abratrary value greater than 1...

I do not want \(\int_{-1}^{1} \int_{-1}^{1} y^z+x^z dydx\)

It is a graph on \(R^2\)

So yeah.... wondering if there is some kind of trick or what... or if the domain of the integral is simply limited to functions

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I wanna take the definite integral of that relation but I guess I am not sure of the notation or if it is even possible...and from x(or/and y?) = -1 to 1 Z is just some abratrary value greater than 1... I do

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So I was just using the textbook answers to check my work but it says to use a graphing utility. Could we go over how to do this on graphing calculators and geogebra possibly?

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There's this awesome thing called THE SECOND FUNDAMENTAL THEOREM OF CALCULUS! I was reading more about the the fundamental theorem of Calculus and the professor mentioned that there was a SECOND. The second theorem basically allows us to create a new function. Honestly, not sure how this helps but it looks cool!

Retrieved from: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/part-b-second-fundamental-theorem-areas-volumes/session-51-the-second-fundamental-theorem-of-calculus/MIT18_01SCF10_Ses51a.pdf