Class Notes: Applications of Integration (Review of Formulas)

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1. Antiderivatives and Integrals

• Derivative Relationship:
  • If ( F'(x) = f(x) ), then ( F(x) ) is called an antiderivative of ( f(x) ).
• Indefinite Integral:
  • ( \int f(x) dx = F(x) + C ), where ( C ) is the constant of integration.
• Integral as Area:
  • The integral sign (( \int )) denotes summing up infinitely small areas.

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2. Rectilinear Motion

• Key Relationships:
  • Velocity is the derivative of position: ( v(t) = s'(t) )
  • Acceleration is the derivative of velocity: ( a(t) = v'(t) )
  • Position is the antiderivative (integral) of velocity
  • Velocity is the integral of acceleration

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3. Definite vs. Indefinite Integrals

• Indefinite Integral:
  • No limits of integration; result is a function
• Definite Integral:
  • Includes lower (( a )) and upper (( b )) limits: ( \int_a^b f(x) dx )
  • Result is a single number

Evaluating Definite Integrals

• Find the antiderivative ( F(x) )
• Plug in upper and lower limits: ( F(b) - F(a) ) (Top minus Bottom)

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4. Properties of Definite Integrals

• ( \int_a^b f(x)dx = -\int_b^a f(x)dx )
• ( \int_a^a f(x)dx = 0 )
• Integral of a constant: ( \int_a^b cdx = c(b-a) )
• Additivity:
  ( \int_a^b f(x)dx + \int_b^c f(x)dx = \int_a^c f(x)dx )
  • Demonstrated by cancellation in the antiderivative evaluation

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5. Area Under Curves and Limit Process

• Definite Integral as Area:
  ( \int_a^b f(x)dx ) gives area under ( f(x) ) from ( x=a ) to ( x=b )
• Riemann Sums:
  ( \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x ), where ( \Delta x = \frac{b-a}{n} )
• Summation Formulas:
  • ( \sum_{i=1}^n C = Cn )
  • ( \sum_{i=1}^n i = \frac{n(n+1)}{2} )
  • ( \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} )
  • ( \sum_{i=1}^n i^3 = \left(\frac{n(n+1)}{2}\right)^2 )
  • (Higher powers with corresponding formulas)

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6. Approximating Area Under a Curve (Riemann Sums and Other Methods)

• Left Endpoints: Use all but last interval point
  • ( \Delta x \sum_{i=0}^{n-1} f(x_i) )
• Right Endpoints: Use all but first interval point
  • ( \Delta x \sum_{i=1}^{n} f(x_i) )
• Midpoint Rule:
  • Use ( x_{i-0.5} ) for the height in each interval
• Trapezoidal Rule:
  • ( T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)] )
• Simpson's Rule:
  • ( S_n = \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n)] )
  • Coefficients alternate: first and last term = 1, middle terms alternate between 4 and 2

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7. The Fundamental Theorem of Calculus (FTC)

• Part 1:
  If ( G(x) = \int_a^x f(t) dt ), then ( G'(x) = f(x) )
• Part 2:
  ( \int_a^b f(x) dx = F(b) - F(a) ), where ( F ) is any antiderivative of ( f )

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8. Net Change Theorem

• Idea:
  ( \int_a^b F'(x) dx = F(b) - F(a) )
• Usage:
  • Accumulation over an interval (e.g., total volume, total displacement, total water supply, etc.)

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9. Area Between Curves

• In terms of ( x ):
  • ( \int_a^b [f(x) - g(x)] dx ) ('top minus bottom')
• In terms of ( y ):
  • ( \int_c^d [f(y) - g(y)] dy ) ('right minus left' curves when integrating with respect to ( y ))

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10. Volumes of Solids of Revolution

• Disc Method (about ( x )-axis):
  • ( V = \pi \int_a^b [r(x)]^2 dx )
• Disc Method (about ( y )-axis):
  • ( V = \pi \int_c^d [r(y)]^2 dy )
• Washer, Shell Methods, and Other Techniques:
  • Formulas follow similar logic, but use appropriate radius and axis orientation

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11. Additional Notes & Resources

• Many more formulas available:
  • Washer method, shell method, cross-sections, work, mean/average value, arc length, surface area, etc.
• Formula sheet and practice problems provided via additional links and resources.
• Simulated and worked-out examples recommended for better understanding.

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End of lesson summary.