# Change of Variables in Double Integrals Examples 1

Recall from the Change of Variables in Double Integrals page that if $z = f(x, y)$ is a continuous two variable real-valued function, $T : S \to R$ is a one-to-one transformation (except possibly on the boundary of $S$), the Jacobian Determinant $\frac{\partial (x, y)}{\partial (u, v)}$ is nonzero, and $x = x(u, v)$, $y = y(u, v)$, and their first partial derivatives with respect to $u$ and $v$ are continuous, then:

(1)

We will now look at some examples of evaluating double integrals using a change of variables.

## Example 1

**Let $z = f(x, y)$ be a a continuous two variable real-valued function and consider the transformation $x = r \cos \theta$ and $y = r \sin \theta$. Use a change of variables to find a formula for $\iint_D f(x, y) \: dA$.**

We must first compute the Jacobian $\frac{\partial (x, y)}{\partial (r, \theta)}$ as follows:

(2)

Therefore $dA = r \: dr \: d \theta$ and so:

(3)

## Example 2

**Evaluate $\iint_D x^2 \: dA$ where $D$ is ellipse $9x^2 + 4y^2 = 36$ and using the transformation $x = 2u$ and $y = 3v$.**

We must first find our new boundary under the transformation given. If we take $9x^2 + 4y^2 = 36$ and plug in $x = 2u$ and $y = 3v$ then:

(4)

Thus out region of integration becomes the unit circle in the $uv$-plane – call this region $D_{uv}$. We can more nicely represent this region with polar coordinates. Let $u = r \cos \theta$ and $v = r \sin \theta$ and then $D_{uv} = \{ (r, \theta) : 0 ≤ r ≤ 1, 0 ≤ \theta ≤ 2\pi \}$.

We now compute the Jacobian $\frac{\partial (x, y)}{\partial (u, v)}$:

(5)

(6)

### 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