Fix the mistake: Integration ② (Definite Integrals, Area, and Volume)

Definition and Calculation of Definite Integrals

A definite integral represents the signed area over the interval [a, b] and can be calculated using the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

When F'(x) = f(x): ∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) − F(a)

Properties of Definite Integrals

  • ∫ₐᵃ f(x)dx = 0
  • ∫ₐᵇ f(x)dx = −∫_b^a f(x)dx
  • ∫ₐᵇ (f+g)dx = ∫ₐᵇ f dx + ∫ₐᵇ g dx
  • ∫ₐᵇ f dx = ∫ₐᶜ f dx + ∫ᶜᵇ f dx
Example ①
Calculate the following definite integrals. (1) ∫₀² (3x²−2x)dx　(2) ∫₋₁¹ (x³−x)dx
Answer:
(1) [x³−x²]₀² = (8−4) − 0 = 4
(2) [x⁴/4−x²/2]₋₁¹ = (1/4−1/2) − (1/4−1/2) = 0 (property of odd functions) = 0
Example ② (Even and Odd Functions)
(1) ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx (f is an even function)　(2) ∫₋ₐᵃ f(x)dx = 0 (f is an odd function)
Calculate ∫₋₂² (x⁴−3x²)dx.
Answer:Since this is an even function (f(−x)=f(x)), = 2∫₀² (x⁴−3x²)dx = 2[x⁵/5−x³]₀² = 2(32/5−8) = 2(−8/5) = −16/5
Key Points
  • Calculate a definite integral as F(b) − F(a) (the constant of integration C disappears)
  • Even function: ∫₋ₐᵃ = 2∫₀ᵃ; odd function: ∫₋ₐᵃ = 0
  • On intervals where f(x) < 0, the sign of the area and the definite integral are opposite

Practice Problems

  1. Calculate ∫₁³ (2x+1)dx.
  2. Calculate ∫₋₁² (x²−x)dx.
  3. Calculate ∫₋₃³ (x⁵−2x³+x)dx.

Answers and Explanations

  1. Answer:[x²+x]₁³ = (9+3)−(1+1) = 10
    Explanation: F(3)=12、F(1)=2。
  2. Answer:[x³/3−x²/2]₋₁² = (8/3−2)−(−1/3−1/2) = 2/3+5/6 = 3/2
    Explanation: F(2)=8/3−2=2/3。F(−1)=−1/3−1/2=−5/6。Difference=2/3+5/6=9/6=3/2。
  3. Answer:0
    Explanation: x⁵−2x³+x is an odd function (each term has an odd degree). ∫₋ₐᵃ(odd function)dx=0。

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