MATH1012 - Lecture 15

← Back to Home

Skeleton Slides

Improper Integrals

• Any integrals that don’t follow the typical ${\int }_a^{b}f(x)d(x)$ format are called improper integrals
  • e.g. where either $a$, $b$ or both are infinity or $f(x)$ is unbounded
  • They can still be interpreted like a normal integral
• Type I improper integral:
  • Integration over an infinite integral
• Type II improper integral:
  • The integral is finite, but the function is unbounded

Type I Improper Integrals

• The improper integral is defined as the limiting value of a suitably chosen definite integral
  • i.e. ${\int }_a^{\infty }f(x)dx=\underset{t\to \infty }{\lim }{\int }_a^{t}f(x)dx$
• If the limit exists, we say the improper integral is convergent and the limit still represents the ‘area’ under the curve
  • Otherwise, we say the improper integral is divergent
• An improper integral where both bounds are infinite is only convergent if both limits are convergent
  • Questions will ask if improper integral is convergent or divergent

Type II Improper Integrals

• Integrals where $a$ or $b$ are a vertical asymptote:
  • ${\int }_a^{b}f(x)dx=\underset{t\to a^{+}}{\lim }{\int }_t^{b}f(x)dx$ or ${\int }_a^{b}f(x)dx=\underset{t\to b^{-}}{\lim }{\int }_a^{t}f(x)dx$
    Where either $f(a)$ or $f(b)$ is not defined
  • If limit exists, the improper integral is convergent
• Integrals where a vertical asymptote exists within the bounds:
  • ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$ where $f(c)$ does not exist
  • Integral is convergent if both $c$ integrals are convergent
    • The right-hand side integrals themselves may need to be solved using combinations of the above techniques
    • If any integral is divergent, the whole integral is divergent

---

MATH1012 - Lecture 16
Annotated Slides