Integration and area under curve

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SUMMARY

Definite integration is fundamentally linked to the area under a curve between a function and the x-axis, as established by the Fundamental Theorem of Calculus. This theorem provides the geometric interpretation that the definite integral calculates the net area, accounting for regions above and below the x-axis. Understanding this concept is essential for students transitioning from indefinite integration to evaluating definite integrals, particularly in calculus courses like AP Calculus.

PREREQUISITES
  • Understanding of indefinite integration
  • Familiarity with summation formulas
  • Basic knowledge of calculus concepts such as limits and continuity
  • Awareness of the Fundamental Theorem of Calculus
NEXT STEPS
  • Study the Fundamental Theorem of Calculus in detail
  • Practice evaluating definite integrals using various functions
  • Explore geometric interpretations of integrals through graphical representations
  • Learn about applications of definite integrals in real-world scenarios
USEFUL FOR

Students in AP Calculus, educators teaching calculus concepts, and anyone seeking to deepen their understanding of the relationship between integration and area under curves.

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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