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Understanding Integration: Indefinite and Definite Integrals, and Fundamental Theorems

This chapter covers the integration process, focusing on indefinite integrals, antiderivatives, and differential equations. Learn how to find particular solutions by solving differential equations and integrating both sides of the equation. Explore the First and Second Fundamental Theorems of Calculus to understand definite integrals and the area under curves. Gain insights into approximation methods like left-sided, right-sided, midpoint, and trapezoid sums. Additionally, discover how to apply the Mean Value Theorem to find average values of functions.

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Understanding Integration: Indefinite and Definite Integrals, and Fundamental Theorems

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  1. Chapter 5 Integration

  2. Indefinite Integral or Antiderivative

  3. Find the Particular Solution or Solve the Differential Equation

  4. Change into a differential equation. Integrate both sides of the equation. Find c by plugging in the coordinate. Replace c and write the particular solution.

  5. 1st Fundamental Theorem of Calculus Definite Integral or Area Under the Curve on the interval [a,b]

  6. Area below the x-axis is NEGATIVE

  7. Approximate the Area Under a Curve Using a Left-Sided Sum

  8. Approximate the Area Under a Curve Using a Right-Sided Sum

  9. Approximate the Area Under a Curve Using a Midpoint Sum

  10. Midpoint Sum

  11. Approximate the Area Under a Curve Using a Trapezoid Sum

  12. Mean Value Theorem (MVT) or Average Value

  13. Mean Value Theorem Average Value

  14. Find the x value where you get the Average Value

  15. Find the Average Value. Set the original function equal to the Average Value. Solve for x.

  16. 2nd Fundamental Theorem of Calculus

  17. 0

  18. Integrate an Even Function

  19. Integrate an Odd Function

  20. U-Substitution or Change of Variables

  21. Find Definite Integral Using U-Substitution or Change of Variables

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