MATH1011 MULTIVARIABLE CALCULUS - WEEK 6 NOTES

MULTIVARIABLE CALCULUS - WEEK 6 NOTES

OVERVIEW

• Topics covered include:
  • Taylor polynomials (two variables)
  • Riemann sums
  • Fundamental Theorem of Calculus (FTC)
  • Applications of Riemann integration
  • Integration techniques

RIEMANN SUMS

Definition

• For a continuous function f(x) on the interval [a, b], the area under the graph of f is denoted as A.
• The approximation of A is given by Riemann sums calculated using partitions:
  • A partition P is defined as:
    P: a = x0 < x1 < x2 < o < x{n-1} < x_n = b
  • Each sub-interval [x{j-1}, xj] has length riangle xj = xj - x_{j-1}.

Riemann Sum Formula

• The Riemann sum for the function f determined by partition P is given by:
  SP = ext{sum from j=1 to n} [f(cj) riangle x_j]
• As partition size ||P|| approaches 0, the sum approaches the exact area:
  A = ext{lim} rac{1}{||P|| o 0} S_P

RIEMANN INTEGRALS

Definition

• A bounded function f is defined as Riemann integrable over [a, b] if:
  ext{lim}{||P|| o 0} ext{sum from j=1 to n} [f(cj) riangle x_j] exists.

Properties

• The integral extstyle rac{a}{b} f(x) du is a real number.
• The definite integral relies on limits, can be computed using uniform partitions, and does not use derivatives or antiderivatives.

FUNDAMENTAL THEOREM OF CALCULUS

• States a relationship between differentiation and integration.
• If F(x) is the antiderivative of f(x):
  • extstyle F'(x) = f(x) for all x ext{ in } I.
  • extstyle A(b) - A(a) = extstyle ext{area} ext{ under } f (integral evaluation):
    extstyle ext{int from a to b} f(x) dx = F(b) - F(a)

APPLICATIONS OF RIEMANN INTEGRATION

Volume by Cross Sections

• To find the volume of a solid with cross-sectional area A(x):
  • V ext{ (volume)} = ext{sum from j=1 to n} A(cj) riangle xj .
  • As ||P|| o 0, V converges to extstyle ext{int from a to b} A(x)dx.

Lengths of Curves

• For a continuously differentiable function f on [a, b]:
• The length of the curve C = ext{set} ext{ of } (x, f(x)): a ext{ to } b is given by:
  L = extstyle ext{int from a to b} ext{sqrt}[1 + (f'(x))^2] dx

INTEGRATION TECHNIQUES

Key Techniques

• Integration by Parts:
  • extstyle ext{int} (uv')dx = uv - ext{int}(u'v)dx
• Integration by Substitution:
  • If g is differentiable:
    extstyle ext{int} f(g(x))g'(x)dx = ext{int} f(u)du with u=g(x).
• Uses half-angle, trigonometric, and inverse trigonometric substitutions.

Numerical Integration

• Techniques include: Midpoint Rule, Trapezoidal Rule, Simpson's Rule for approximating integrals, especially when exact forms are hard to find.