Can you explain what one would do if the horizontal line being sought happened to be both over AND under the curve on the interval of integration. As opposed to the easy case here of having a horizontal line only under the curve x^2 on the whole interval of integration.
I mean...can you give me a specific problem? You must have one in mind. I have several ideas of what you're potentially talking about. In all of them I'd pretty much let the line be y=k, I'd try to find the intersections with the curve (3 of them, x-coords) in terms of k, and then I'd set up two integrals that have all four bounds in terms of k and the integrand in terms of k. I'm guessing solving that would be very difficult (if not impossible in almost every case) by hand.
@@turksvids hi turksvids thanks for getting back. Basically the problem necessarily involves not being able to say if y=k stays under the curve or if y=k goes both under and over the curve. Because we don't know k for starters obviously; were trying to find a k that divides the area in two. So for example if you have a curve with the leading term like x^3 or x^5, not a neat x^2 curve where you know y=k would be under the curve on the interval. Essentially where you just don't know if the dividing line y=k stays under the top curve or not on the interval (because we don't know k a priori). I was surprised to find that this was harder than I thought (and i don't know how to do it) since any definite integral gives the net signed area as you know, so you're left wondering if the curve dips below your y=k on the interval and you have to account for the negative bit somehow. I'm still stuck on it and I'm surprised that the definite integral is seemingly powerless in finding area in this way. An example that might illustrate what I'm talking about is: find a line y=a that divides the area under the curve 2x^3-x^2-x+1 from 0 to 1 into 2 equal areas. As long as we have the issue of not knowing if the curve dips below the y=k line in the rightmost part of the interval, this illustrates what I'm wondering about.
That's the integrand that results from finding top - bottom, not the function you graph to find the region. The functions you graph to find the region are y = x^2 and y = 9, where y = 9 is on top, so top - bottom is 9 - x^2. Hope this helps!
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Can you explain what one would do if the horizontal line being sought happened to be both over AND under the curve on the interval of integration. As opposed to the easy case here of having a horizontal line only under the curve x^2 on the whole interval of integration.
I mean...can you give me a specific problem? You must have one in mind. I have several ideas of what you're potentially talking about. In all of them I'd pretty much let the line be y=k, I'd try to find the intersections with the curve (3 of them, x-coords) in terms of k, and then I'd set up two integrals that have all four bounds in terms of k and the integrand in terms of k. I'm guessing solving that would be very difficult (if not impossible in almost every case) by hand.
@@turksvids hi turksvids thanks for getting back. Basically the problem necessarily involves not being able to say if y=k stays under the curve or if y=k goes both under and over the curve. Because we don't know k for starters obviously; were trying to find a k that divides the area in two. So for example if you have a curve with the leading term like x^3 or x^5, not a neat x^2 curve where you know y=k would be under the curve on the interval. Essentially where you just don't know if the dividing line y=k stays under the top curve or not on the interval (because we don't know k a priori).
I was surprised to find that this was harder than I thought (and i don't know how to do it) since any definite integral gives the net signed area as you know, so you're left wondering if the curve dips below your y=k on the interval and you have to account for the negative bit somehow.
I'm still stuck on it and I'm surprised that the definite integral is seemingly powerless in finding area in this way.
An example that might illustrate what I'm talking about is: find a line y=a that divides the area under the curve 2x^3-x^2-x+1 from 0 to 1 into 2 equal areas. As long as we have the issue of not knowing if the curve dips below the y=k line in the rightmost part of the interval, this illustrates what I'm wondering about.
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If it is 9-x^2, shouldn't the parabola be inverted with the vertex at (0,9)?
That's the integrand that results from finding top - bottom, not the function you graph to find the region. The functions you graph to find the region are y = x^2 and y = 9, where y = 9 is on top, so top - bottom is 9 - x^2. Hope this helps!
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