Chapter 13 - Intergration

    Cards (19)

    • What symbol represents the process of integration?
    • What does the equation ∫f′(x)dx = f(x) + c represent?
      It represents the indefinite integral of f′(x)
    • How is the process of integrating xⁿ expressed?
      ∫xⁿdx = xn+1n+1+\frac{x^{n+1}}{n+1} +c c, n ≠ -1
    • What does the dx in the integral signify?
      It indicates integration with respect to x
    • How do you integrate a polynomial function?
      Integrate each term one at a time
    • What is the rule for integrating the sum of two functions?
      [f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
    • What should you not do with the constant term c when integrating?
      Do not multiply it by k
    • What is the result of integrating dy/dx = kxⁿ?
      y = kn+1xn+1+\frac{k}{n+1}x^{n+1} +c c, n ≠ -1
    • How is the function notation expressed for f′(x) = kxⁿ?
      f(x) = kn+1xn+1+\frac{k}{n+1}x^{n+1} +c c, n ≠ -1
    • Why can't you use the integration rule if n = -1?
      Because 1/(n+1) = 1/0 is undefined
    • What are the steps to find the constant of integration c?
      Integrate, substitute values, solve for c
    • What is a definite integral?
      An integral calculated between two limits
    • What does a definite integral usually produce?
      A value
    • What are the three stages of calculating a definite integral?
      Write statement, integrate, evaluate f(b) - f(a)
    • How can definite integration be used with areas under curves?
      To find the area under a curve
    • What is the formula for the area under a positive curve?
      Area = ∫ᵇₐ y dx
    • What happens when the area bounded by a curve is below the x-axis?
      ∫ᵇₐ y dx gives a negative answer
    • How can definite integration be combined with other geometric shapes?
      To find complicated areas on graphs
    • What are the key points summarized in Chapter 13?
      1. If dy/dx = xⁿ, then y = 1n+1xn+1+\frac{1}{n+1}x^{n+1} +c c, n ≠ -1.
      2. If dy/dx = kxⁿ, then y = kn+1xn+1+\frac{k}{n+1}x^{n+1} +c c, n ≠ -1.
      3. ∫f′(x)dx = f(x) + c
      4. ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
      5. To find c: integrate, substitute values, solve.
      6. Definite integral: ∫ᵇₐf′(x)dx = f(b) - f(a).
      7. Area = ∫ᵇₐ y dx for positive curves.
      8. Negative area for curves below x-axis.
      9. Combine with trapeziums and triangles for complex areas.
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