Applications of Riemann Sum

\(\int_{a}^b f(t)dt\)= lim_(n->∞)\(\sum_{k=1}^{n} f(t) Δ t\)

=lim_(n->∞)\(\sum_{k=1}^{n} f(a + (\frac{b-a}{n})k*(\frac{b-a}{n})\)

\(\int_{1}^5 x^3dx\) = lim_{(n-> ∞)}\(\sum_{k=1}^{n} (1 + (\frac{5-1}{n}{)k)}^3(\frac{5-1}{n} )\)

= lim_{(n-> ∞)}\(\sum_{k=1}^{n} (1 + (\frac{4}{n}{)k)}^{3*}(\frac{4}{n} )\)

Numerical methods for Integration

• The most common techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson’s rule.
• The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.

Equations  are :

• Midpoint rule
  M_n= lim_(n->∞)\(\sum_{k=1}^{n} f(m_i) t\)
• Trapezoidal rule

T_n=\(\frac{x}{2}\)(f(x₀)+2f(x₁)+2f(x₂)……..2f(x_n-1)+f(x_n))

• Simpson’s rule

S_n=\(\frac{x}{3}\)(f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+2f(x₄)….+2f(x_n-2)+4f(x_n-1)+f(x_n)

The Area Function

Define \(\int_{a}^x f(t)dt\)  as the area A  of the domain D  formed by the curve of f(t), the x-axis and the lines x=a and x=b.

Areas and Antiderivatives

F(x)=\(\int_{a}^b f(t)dt\)

Definite Integrals Fundamental Theorem of Calculus

F(b)-F(a)=[F(t)]a^b

\(\int_{a}^b f(t)dt\) = F(t)a^b =  F(b) – F(a)

\(\int_{a}^b f(t)dt\) + \(\int_{b}^c f(t)dt\)  = \(\int_{c}^a f(t)dt\)  (Chasles’ Relation)

\(\int_{a}^a f(t)dt\) =0

\(\int_{b}^a f(t)dt\) =-\(\int_{a}^b f(t)dt\)

\(\int_{a}^b (f+g)(t)dt\) = \(\int_{a}^b f(t)dt\) +\(\int_{a}^b g(t)dt\)

\(\int_{a}^b kf(t)dt\) = k \(\int_{a}^b f(t)dt\)

Special integrals

\(\int_{a}^b |x|dx\) = \(\int_{a}^c -xdx\)  +  \(\int_{c}^b xdx\)

Area and Definite Integrals

Miscellaneous Applications

If a > b, the mean of the function y = f(x)  on the Interval = [a,b]  is

\(\bar{y}\) = \(\frac{1}{b-a}\) \(\int_{a}^b f(x)dx\)

Techniques of Integration

By substitution

See the chapter on trigonometric substitution.

Integration by parts

\(\int_{a}^b\)u’(x)v(x)dx = [u(x)v(x)]_a^b–\(\int_{a}^b\)u(x)v’(x)dx