# Improper Double Integrals

Just like in single variable calculus, we can sometimes encounter improper double integrals if we integrate a two variable real-valued function if our domain of integration is unbounded or if the function we are integrating is unbounded for some point in our domain.

We will only be dealing with improper integrals for which $f(x, y) ≥ 0, \forall (x, y) \in D$, or $f(x, y) ≤ 0, \forall (x, y) \in D$. If $f$ changes sign on $D$, then improper double integrals are much more complicated to evaluate, and their iterated integrals my converge when in actuality their double integral does not.

That said, evaluating improper double integrals of the simpler type specified above is much like evaluating regular improper integrals. For example, suppose that wanted to evaluate the integral $\iint_R \frac{1}{(1 + x^2)(1 + y^2)} \: dA$ where $R = \{ (x, y) : x ≥ 0, y ≥ 0 \}$. If we rewrite this double integral as iterated integrals, we would have that:

(1)

If we evaluated in the inner improper integral $\int_0^{\infty} \frac{1}{(1 + x^2)(1 + y^2)} \: dy$ first while treating $x$ as fixed, we would get that:

(2)

Therefore we have that:

(3)

Therefore, $\iint_R \frac{1}{(1 + x^2)(1 + y^2)} \: dA$ converges to $\frac{\pi^2}{4}$. The graph of $z = \frac{1}{(1 + x^2)(1 + y^2)}$ in the first quadrant is given below:

### Related post:

- Grade Nine learners taught mathematics skills – Tembisan
- A Library Browse Leads Math’s Bill Dunham to Question the Origins of The Möbius Function – Bryn Mawr Now
- Year 5 and 6 students to sit competition this Wednesday – Great Lakes Advocate
- USC student wins silver medal in China math contest – SunStar Philippines
- CBSE Exam 2020: Two separate examinations to be conducted for Class 10 Mathematics – Jagran Josh
- Concepts incomplete, problems unsolvable in math textbooks – Times of India
- Education Ministry to Host Tertiary and Employment Fairs – Government of Jamaica, Jamaica Information Service
- Vogue’s Edwina McCann and Westpac’s Anastasia Cammaroto on how they inspire women to pursue STEM – Vogue Australia
- Jonee Wilson, Temple Walkowiak to Measure High-Quality Instructional Practices to Support Marginalized Students in Rigorous Mathematics through NSF Grant – NC State College of Education
- Australian Conference on Science and Mathematics Education – Australian Academy of Science