If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Integral Calculus>Unit 1

Lesson 20: Proof 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.

## Want to join the conversation?

• 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.
• 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.
• Please point to the best calculator online , for these equations .. Thanks
• I've found that symbolab is amazing for all levels of math
• 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.
• 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?
• I don't understand how to solve problems by this theory explained
• I can relate to your position. As you keep getting higher and higher in math, the amount of "solution by application of rules and formulas" starts to decrease, and the "solution by application of implications and consequences of theorems" starts to increase. In a first year calculus program, that usually happens first with the formal definition of limits, and then again here at the fundamental theorem of calculus. Here is a brief explanation of the relation of the F.T.o.C. and the type of problem you may encounter testing your insight/understanding of the theorem. I hope it helps. If not, do not hesitate to ask for clarification. I like to take breaks from my regular work and come here to help students, so I will answer as fast as I can. Here is the link: http://bajasound.com/khan/khan0002.jpg.
Keep studying!