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### Course: AP®︎/College Calculus BC>Unit 6

Lesson 16: Optional videos

# Intuition for second part of fundamental theorem of calculus

The second part of the fundamental theorem of calculus tells us that to find the definite integral of a function ƒ from 𝘢 to 𝘣, we need to take an antiderivative of ƒ, call it 𝘍, and calculate 𝘍(𝘣)-𝘍(𝘢). Get some intuition into why this is true. Created by Sal Khan.

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• i dont entirely understand how finding the area under the line of v(t) would give u the change in position if change in position is equal to change in position/t
• v(t) is just a function of velocity with respect to time. that means that velocity will be the dependent variable (y) and time will be the independent variable (x). since velocity is the same thing as distance/time we can rewrite the y variable as distance/time. Therefore, when we find the area under the v(t) function, we are multiplying the combination of a bunch of rectangles together with the base being time (x) and the height being velocity (y or distance/time). Therefore, when we do the following calculation: time*(distance/time), the 'time's cancel out and we are left with just distance travelled which is the same thing as change in position.
• Please point to the best calculator online , for these equations .. Thanks
• I've found that symbolab is amazing for all levels of math
• At , I thought that this was First fundamental theorem of calculus (basically, how to take an integral).
I learned that the second fundamental theorem of calculus was: that if you take the derivate of an integral from 0 to x, you get f(x). Is this incorrect?
• Yes, that is what my calculus book says too. Farther down on the playlist for "indefinite and definite integrals" is a set of videos for "fundamental theorem of calculus". In these videos it becomes clear what Sal's designation of "first" and "second" is just switched of what our book says. Another comment in those section of videos says that in other references the first and second theorems of calculus are just parts of one theorem of calculus. I don't think it matters which you view as first and which you view as second. They are really just different viewpoints of the same idea.
• At , Sal says F(x) is 'an' anti-derivative of f(x). How can a single curve have multiple anti-derivatives?
• When you take the derivative, if there is a constant term, it disappears. so you're losing information. When you find the antiderivative, you don't know what the constant term was, which is why a single curve has (infinitely) many antiderivatives.
• Is the area under the upper graph useful for anything?
It looks like it would be a Reimann sum of delta-t * s(t-1), which is a sum of times * distances, or in other words the sum of velocities at specific moments. And that doesn't make much sense.
• Thinking of it in measurements, the unit of the area under the first curve would be m*s (position times time) and that certainly doesn't make sense. Abstractly speaking though, area under the curve of a function (for example f(x) ) can describe an aspect of of it's anti-derivative (F(x) ) so long as it has one.
• Why the capital F notation?
• In the beginning I always thought of the antiderivative as the Original Function, so if I saw f(x) I would think derivative and F the anti or original function.
• My teacher taught this as the first fundamental theorem of calculus, not second.
• I believe most authors use the terminology your teacher uses. Not a big deal, though: it's basically an arbitrary choice as to which one you call first and which you call second. You just have to be aware of which one Sal (or your teacher) is talking about.
• Is this the first fundamental theorem? My book has the second fundamental theorem as being d/dx integral on [x,a] of f(t)dt=f(x). Where can you find the video on the second fundamental theorem?