Stuck on an Area Between Curves Question

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SUMMARY

The discussion centers on calculating the area between the curves defined by the functions f(x) = x³ - 9x² + 18x and g(x) = -x³ + 9x² - 18x. The user initially found the points of intersection at x = 0, 3, and 6, and attempted to compute the area by integrating |f(x) - g(x)| over the interval [0, 6]. Despite obtaining a result of zero, the user realized the importance of correctly applying absolute value signs in the integration process, which led to the conclusion that the area calculation was incorrect due to this oversight.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration and area between curves.
  • Familiarity with polynomial functions and their properties.
  • Knowledge of absolute value functions and their implications in integration.
  • Experience using graphing calculators for visualizing functions.
NEXT STEPS
  • Review the process of finding the area between curves using definite integrals.
  • Study the application of absolute value in integration, particularly in cases of intersecting functions.
  • Practice solving similar problems involving polynomial functions and their intersections.
  • Explore advanced graphing calculator techniques for verifying integration results visually.
USEFUL FOR

Students and educators in calculus, mathematicians focusing on integral calculus, and anyone seeking to improve their skills in finding areas between curves.

Camronnba
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I have been asked to find the area between the following curves
f(x)= x^3 -9x^2 +18x and
g(x)= (-x)^3 +9x^2 -18x

I started out by finding the points of intersection, which I found to be 0, 3, and 6. I then integrated |f(x)-g(x)|and evaluated between 6 and 0. I got an answer of zero but it says I am wrong. I then tried evaluating between 0 and 3, and 3 and 6 and adding those together, again I get zero. After that I brought out my graphing calculator, and after viewing the graphs, 0 seems like a logical answer. I must be making a mistake somewhere, if someone could please steer me in the right direction it would be greatly appreciated. Thanks
 
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Nevermind, I should really pay attention to absolute value signs when I see them. haha
 

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