Confusion about the constant of integration and bounds

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SUMMARY

The discussion clarifies the concept of the constant of integration in the context of definite and indefinite integrals. When integrating a derivative from bounds x1 to x2, the constant of integration is not included in the final result, as it would alter the area calculation beneath the curve. The constant C represents an infinite number of solutions for indefinite integrals, but for definite integrals, it is omitted to maintain accuracy in area representation.

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  • Familiarity with definite and indefinite integrals.
  • Knowledge of the properties of integrals and their geometric interpretations.
  • Basic grasp of the concept of area under a curve.
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  • Study the Fundamental Theorem of Calculus to understand the relationship between differentiation and integration.
  • Explore examples of definite integrals to see how constants of integration are handled.
  • Learn about the geometric interpretation of integrals, focusing on area calculations.
  • Review common integration techniques and their applications in solving real-world problems.
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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of integration concepts and their applications in problem-solving.

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When we integrate a derivative:
gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif

What are the bounds?
gif.latex?\int_{?}^{?}dy=&space;\int_{?}^{?}x^{5}+x^{4}+x^{3}dx.gif

Shouldn't we then have this if the bounds are x2 and x1:
}{6}+\frac{x_{1}^{5}}{5}+\frac{x_{1}^{4}}{4}&space;\right&space;]&space;+&space;C.gif

but instead we have this:
f.latex?y=&space;\frac{x_{_{2}}^{6}}{6}+\frac{x_{2}^{5}}{5}+\frac{x_{2}^{4}}{4}+C.gif
 

Attachments

  • gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif
    gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif
    589 bytes · Views: 475
Physics news on Phys.org
The constant of integration is a way of acknowledging that there are infinitely many solutions for every indefinite integral. When we integrate any function from [tex]x_1[/tex] to [tex]x_2[/tex] we are finding the area beneath the curve. Adding on a constant C would no longer give us the correct area beneath the curve, provided that C is not equal to zero.
 

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