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### Course: Integral Calculus>Unit 3

Lesson 4: Area: vertical area between curves

# Area between a curve and the x-axis

Based on the fundamental theorem of calculus, we can use antiderivatives to compute integrals. Created by Sal Khan.

## Want to join the conversation?

• I am trying to jive in my head how the mandatory +c (constant) after antiderivatives of functions doesn't matter for definite integrals.
Hmm.
• I'm seven years late, but it's because it cancels out.
s stands for the long s

` b s f(x) = (F(a) + C) - (F(b) + C) a`
Note that while we sometimes treat C as an arbitrary constant, here we're using a specific antiderivative of f(x), and we're using the same one; so the C's are the same, and can cancel out.
• What is the point of Riemann approximation when we have the Second Fundamental Theorem of Calculus?
• Also, if you ever go into computer programming, it is much easier to use a really precise Riemann sum instead of actually integrating. This is because we can't make the programs flexible or intelligent enough to handle all forms of integrals. (As a side-note, this is why most graphing calculators can only do definite integrals; they can't do the actual integral, only find the area)
• So basically, the antiderivative of a function at "x" tells you the area from 0 to x under the curve?
• Yes, that is exactly correct. Incredibly useful, isn't it?
• In all of my life experience, I have only ever been concerned with f(x) (the curve itself). What exactly is the value in knowing the area under a curve?
• Think of it this way
You're an engineer/architect. I'm your boss. I tell you: "I want an arch bridge that looks awesome. Give me one."
So you create blueprints for a bridge. But how would you know how much cement to use? Let's say the equation for your curve is y = -x²/1000 + 10. You can use integral calculus to find the amount of cement you will need.
If you are a statistician, you will need to find the area of a Gaussian curve more than once. Its equation: ƒ(x) = ae^((x-b)²/-2c²).
If you are counting an infinite series (which comes up a lot), the area under the curve is almost exactly the answer.
If anyone else wants to add a couple other reasons, they can.
Hope this helps!
• Did we not learn this before? I'm so confused, why are we learning the most basic stuff in the first unit of integral calculus.
• why it is called "definite" integral?
• In a definite integral the region of ingratiation is defined. The integral from a to b means that we are integrating from a to b which is a definite length.

In an indefinite integral the region of integration is not defined.
• I don't get the explanation from the "Area between a curve and an axis" exercises. https://puu.sh/mQCj2/e83d91d76e.png Can someone explain it in a not-too-rigorous way?
• Make sure you fully understand the differential calculus lessons or learning integration will be more difficult.

Here is a great introduction to integration and area under the curve.
https://www.mathsisfun.com/calculus/integration-introduction.html

If you still have any specific questions after going through the math is fun page, write them in the comments and I will reply.
Keep Studying and Have Fun!
• at why does sal take the derivative of x^3?
• It s true that it is easy if you have also watched the videos of how to take anti-derivatives. But if one is only watcing the videos of definite integrals, he just might not know how to take the anti-derivate of a function.
• what is the difference between definite integral and indefinite integral?
• Another thing that might help. The answer to an indefinite integral is a function. The answer to a definite integral is a value, a number. For example, in the problem for this video, the indefinite integral is (1/3)x^3 + c. The definite integral, evaluated from 1 to 4 is 21. You use the indefinite integral to find the definite integral evaluated between two values.