Question

Find the area of the domain D which is in the first quadrant and bounded by xy=1, xy=3, y=x, y=3x.

Answer 1

solve for y, and draw a picture with all the curves...

Answer 2

Is there a way to do this without sketching?

Answer 3

unless you have a great mind for imagining how a closed bouded region will look like....I dont

Answer 4

bounded

Answer 5

Let's use a change of variables \(u=xy,v=\frac{y}x\) to get new bounds $$u=1,u=3,v=1,v=3$$Now, we compute our Jacobian:$$\det\begin{bmatrix}u_x&u_y\\v_x&v_y\end{bmatrix}=\det\begin{bmatrix}y&x\\-\frac{y}{x^2}&\frac1x\end{bmatrix}=\frac{y}x+\frac{xy}{x^2}=2\frac{y}{x}=2v$$
Now throw it all together:$$\iint_DdA=\int_1^3\int_1^3(2v)\,du\,dv=\int_1^32v\int_1^3du\,dv=4\int_1^3dv=8$$