The thing that always screws up my head about such explanations is that AREA is measured in units squared, but you say the areas is not in units squared. It's like saying the area of my living room is 432 - with no units. I could never get past that quandry.
The units in an integral problem are square units but they are implied. The area that they are talking about is the area under a curve. The curve is described by a function f(x). On a graph f(x) represents the height of the curve above a point x on the horizontal axis. The dx that appears in the integral is an infinitesimal length, almost zero but not quite, along the horizontal axis. When you see f(x)·dx it represents a height times width: the area of a rectangle, an infinitesimally thin rectangle. In the integral you're taking the areas of all such infinitesimally thin adjacent rectangles and adding them up. Since there is no space between one rectangle and the next the sum total is the exact area under the curve.
@unicyclechinese Although you feel so, you’re not in a quandary. You’re correct in your understanding. The area is square units. It’s the instructor’s fault. This instructor is too sloppy and careless much too often, especially seeing that he’s teaching. Area is always given in square units and always understood to be in square units. Just as volume is always given in cubic units and always understood to be in cubic units. Just that, because that’s understood, the units aren’t written, or spoken, oftentimes, is all. It goes without saying, sort of speak. Bad habit, though, when doing more complex calculations because that leads to errors in the units portion as well as the numerical value of answers. Takes a little more time including units sometimes, depending on the level of complexity of the problem, but doing so allows for less error. But you’re correct and absolutely right, the area is 7.5 square units, here! And everywhere else in science and the world.
Well, if I can assume that the angle of the sides 3 and 5, is a right angle despite it not being marked, the area is 3×5÷2=7.5. At least that's my answer.
I suspect this calculus example was particularly easy because the function (read: the line) was linear, which made 'de-integration' easy. But the curve you drew late in the video was at least a cubic equation. How do you even find that equation in the first place? Do you start by averaging the minima and maxima and use the result to lower the curve to center it around the X axis, then find its zero intercepts? I'm just guessing wildly here.
It is impossible to find the function based on the graph. The graph will never give you the exact values of the function. Even if you give the extreme values and the "zeros", you still don't know the values for the other x's.
I needed help both ways. It was a great review. I try to do one or two problems a day. It’s great! Thanks
I like this one. Sill don't get how integrating adds up all those little rectangles, but guessig it has to do with the tangent line. :)
Lot of thanks.💗
Vary nice explanation...... ❤️
7.5 square units By geometry and by the calculus it gets 45/6 which reduces to 7.5 square units.
The thing that always screws up my head about such explanations is that AREA is measured in units squared, but you say the areas is not in units squared. It's like saying the area of my living room is 432 - with no units. I could never get past that quandry.
The units in an integral problem are square units but they are implied. The area that they are talking about is the area under a curve. The curve is described by a function f(x). On a graph f(x) represents the height of the curve above a point x on the horizontal axis. The dx that appears in the integral is an infinitesimal length, almost zero but not quite, along the horizontal axis. When you see f(x)·dx it represents a height times width: the area of a rectangle, an infinitesimally thin rectangle. In the integral you're taking the areas of all such infinitesimally thin adjacent rectangles and adding them up. Since there is no space between one rectangle and the next the sum total is the exact area under the curve.
@@johnnolen8338 Excellent explaination.
@unicyclechinese
Although you feel so, you’re not in a quandary. You’re correct in your understanding. The area is square units. It’s the instructor’s fault. This instructor is too sloppy and careless much too often, especially seeing that he’s teaching. Area is always given in square units and always understood to be in square units. Just as volume is always given in cubic units and always understood to be in cubic units. Just that, because that’s understood, the units aren’t written, or spoken, oftentimes, is all. It goes without saying, sort of speak. Bad habit, though, when doing more complex calculations because that leads to errors in the units portion as well as the numerical value of answers. Takes a little more time including units sometimes, depending on the level of complexity of the problem, but doing so allows for less error. But you’re correct and absolutely right, the area is 7.5 square units, here! And everywhere else in science and the world.
1/2 of 15 is 7.5 sq. Unit
Nice to revisit this after so many years out of school.
Well, if I can assume that the angle of the sides 3 and 5, is a right angle despite it not being marked, the area is 3×5÷2=7.5.
At least that's my answer.
Thats how I figured it.. Geometry wins.
I suspect this calculus example was particularly easy because the function (read: the line) was linear, which made 'de-integration' easy. But the curve you drew late in the video was at least a cubic equation. How do you even find that equation in the first place? Do you start by averaging the minima and maxima and use the result to lower the curve to center it around the X axis, then find its zero intercepts? I'm just guessing wildly here.
It is impossible to find the function based on the graph. The graph will never give you the exact values of the function. Even if you give the extreme values and the "zeros", you still don't know the values for the other x's.
Calculus 1
But not mindblowing