Understanding Integration: A Simple Explanation

  • Thread starter Thread starter ozone
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary
SUMMARY

This discussion focuses on the process of integrating a function with respect to 'x', specifically using the anti-derivative function a²x - (1/3)x³. The key steps involve careful handling of minus signs and evaluating the anti-derivative at the limits x = a and x = -a, or alternatively at x = a and doubling the result due to the symmetry of the integrand about the y-axis. The conclusion emphasizes the importance of understanding the behavior of the function at these critical points.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly integration.
  • Familiarity with anti-derivatives and their properties.
  • Knowledge of evaluating definite integrals.
  • Concept of symmetry in mathematical functions.
NEXT STEPS
  • Study the properties of anti-derivatives in calculus.
  • Learn about definite integrals and their evaluation techniques.
  • Explore the concept of function symmetry and its implications in integration.
  • Practice integration problems involving constants and variable limits.
USEFUL FOR

Students of calculus, mathematics educators, and anyone looking to deepen their understanding of integration techniques and anti-derivative evaluation.

ozone
Messages
121
Reaction score
0
If one could just explain how the attached answer below was derived I would be greatful. Thanks!
 

Attachments

  • eq1.png
    eq1.png
    1.1 KB · Views: 424
Physics news on Phys.org
By being very careful with your minus signs! What did you try? Remember that 'a' is a constant and you are integrating with respect to 'x'.
 
The anti-derivative function is a^{2}x - \frac{1}{3} x^{3} . You then evaluate it at x = a and x = -a (or at x = a and double the result, since the integrand function is symmetrical about the y-axis [the anti-derivative is zero at x = 0 ] ).
 

Similar threads

Solving a linear differential equation with constant coefficients
  • · Replies 9 ·
Replies
9
Views
2K
Notion of an integral as a summation
  • · Replies 4 ·
Replies
4
Views
1K
How integral and gradient cancels?
  • · Replies 13 ·
Replies
13
Views
2K
Two ways of integration giving different results
  • · Replies 9 ·
Replies
9
Views
2K
Integration Substitution Techniques for quadratic expressions under square roots
  • · Replies 9 ·
Replies
9
Views
2K
Partial Differentiation of this Equation in x and y
  • · Replies 4 ·
Replies
4
Views
899
Integral of e^cosx: Answers Sought
  • · Replies 11 ·
Replies
11
Views
3K
Changing the interval of integration
  • · Replies 2 ·
Replies
2
Views
1K
Area surface of revolution (rotating this astroid curve around the x-axis)
  • · Replies 2 ·
Replies
2
Views
2K
Obtaining the moment of inertia of a sphere and spherical shell
  • · Replies 6 ·
Replies
6
Views
1K