Integration and area under curve

In summary, the conversation discusses the concept of definite integration and its relationship to the area under a curve. The speaker expresses confusion about how definite integration, which they have learned about in their calculus class, can equal the area under a curve. They wonder if their lack of understanding is due to not yet learning how to evaluate definite integrals. A link to the Fundamental Theorem of Calculus is provided for further explanation.
  • #1
jmsdg7
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I have seen a thread with a similair title but passed up on what i want to know.

I just want somebody to explain to me why definite integration equals the area under between the function and the x axis

Ive just been through indefinite integration, then using the summation formulas in my AP calculus class, and i understand the concept of going from acceleration to velocity to position, but i don't understand how that can equal the area under a curve.

this whole question might be pointless because we haven't learned how to evaluate definite integrals yet and maybe my answer lies in that.

Any 2 cents worth is appreciated!
 
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1. What is integration and how is it related to the area under a curve?

Integration is a mathematical concept that involves finding the sum of infinitely small parts of a given function. The area under a curve is a specific type of integration that calculates the total area between a curve and the x-axis.

2. Why is finding the area under a curve important?

Finding the area under a curve is important because it allows us to solve a variety of real-world problems, such as calculating distances, volumes, and probabilities. It also helps us understand the behavior of a function and make predictions about its values.

3. What are the different methods for finding the area under a curve?

The two main methods for finding the area under a curve are the Riemann sum and the definite integral. The Riemann sum involves dividing the area into smaller rectangles and summing their areas, while the definite integral uses calculus to find the exact area under the curve.

4. Can the area under a curve be negative?

Yes, the area under a curve can be negative if the curve dips below the x-axis. In this case, the negative area represents the amount of space between the curve and the x-axis, and the positive area represents the area above the x-axis.

5. How is the area under a curve related to the antiderivative?

The area under a curve is related to the antiderivative through the Fundamental Theorem of Calculus, which states that the definite integral of a function is equal to the difference between the antiderivatives of its upper and lower limits of integration. This allows us to use antiderivatives to easily find the area under a curve without needing to use geometric methods.

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