Applications of Integration: Lecture Notes

Indefinite and Definite Integrals

• Antiderivatives

  • The derivative of capital F is the same as lowercase f.
  • The antiderivative of lowercase f is capital F.
  • Indefinite integral of f(x) gives the antiderivative (capital F) plus constant C.

• Rectilinear Motion

  • Velocity function is the derivative of the position function.
  • Instantaneous acceleration function is the derivative of the velocity function.
  • Integral/antiderivative of the velocity function gives the position function.
  • Antiderivative/indefinite integral of acceleration function gives velocity function.

Definite Integrals

• Concept

  • Indefinite integrals lack limits of integration; definite integrals have limits.
  • Result of a definite integral is a number; result of an indefinite integral is a function.

• Evaluation

  • To evaluate a definite integral, find the antiderivative F, then plug in the upper and lower limits: F(b) - F(a).

• Properties

  • Swapping limits changes sign: (\int_a^b f(x) dx = -\int_b^a f(x) dx).
  • Integral over a point (a to a): (\int_a^a f(x) dx = 0).
  • Integral of a constant: (C(b-a)).
  • Additive property: (\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx).

Calculating Area Under a Curve

• Evaluate the definite integral of the function.
• Use a limit process with Riemann sums if the function is complex or undefined.
  • Break into rectangles, calculate area of each using height (f(x)) and width (Δx).
  • As n (number of rectangles) approaches infinity, the approximation approaches the exact area.

Approximation Methods

• Left Endpoints

  • Use all initial points except the last.

• Right Endpoints

  • Use all points from second to last.

• Midpoint Rule

  • Use middle points.

• Trapezoidal Rule

  • Formula: (T_n = \frac{Δx}{2} [f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]).

• Simpson’s Rule

  • Formula: (S_n = \frac{Δx}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)]).

Fundamental Theorem of Calculus

• 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)).

Net Change Theorem

• Similar to FTC part 2.
• Integral of the derivative of a function gives net change: (\int_a^b F'(x) dx = F(b) - F(a)).

Area Between Curves

• In terms of x: (\int_a^b [f(x) - g(x)] dx).
• In terms of y: (\int_c^d [f(y) - g(y)] dy).

Volume of Solids of Revolution

• Disk Method

  • Formula for rotation about x-axis: (V = \pi \int_a^b [radius(x)]^2 dx).
  • For y-axis: (V = \pi \int_c^d [radius(y)]^2 dy).

• Additional Methods

  • Washer method, shell method, volume by cross-sections are also possible.

Additional Resources

• Formula sheets and additional example problems will be provided in links for further practice.