Q. 48 - Calculus

Question

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.  

$\int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA$, where $\Omega$ is the region from Exercise 47.

Step-by-Step Solution

Answer

$\int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\left(\frac{3}{4}\ln \left(\frac{4}{3}\right)-\frac{3}{8}\right)$

Step 1: Draw the region and name the vertices

The region $\Omega$is bounded by,

$x=0, y^2-x^2=4, y=2x, y^2-x^2=1$

Plot the given points to form the region and name the vertices.

In the above region, the equations of boundary curves are,

$$\begin{gathered} AB: y^2-x^2=1 \\ BC: y=2x \\ CD: y^2-x^2=4 \\ DA: x=0 \end{gathered}$$

Consider the new set of variables defined as,

$$\begin{gathered} u=y^2-x^2 \\ v=\frac{x}{y} \end{gathered}$$

After solving we get that,

$$\begin{gathered} \sqrt{\frac{u}{1-v^2}}=y \\ \sqrt{\frac{u{v}^2}{1-v^2}}=x \end{gathered}$$

Step 2: Determine the equation of each boundary in terms of u and v.

We have,

$$\begin{gathered} \sqrt{\frac{u}{1-v^2}}=y \\ \sqrt{\frac{u{v}^2}{1-v^2}}=x \end{gathered}$$

Use these equations to determine the equation of each boundary of the region. 

$$\begin{gathered} AB: y^2-x^2=1\Rightarrow u=1 \\ BC: y=2x\Rightarrow v=\frac{1}{2} \\ CD: y^2-x^2=4\Rightarrow u=4 \\ DA: x=0\Rightarrow v=0 \end{gathered}$$

Plot these limits on u v plane.

Step 3: Evaluate the double integral.

Set up the double integral,

$$\begin{gathered} \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\frac{1}{4}{\int }_{u=1}^{u=4}{\int }_{v=0}^{v=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\frac{1}{1-v^2}-1\right)\left(2v\right) dvdu \\ \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\frac{1}{4}{\int }_{u=1}^{u=4}{\int }_{v=0}^{v=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\frac{2v}{1-v^2}-2v\right) dvdu \\ \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\frac{1}{4}{\int }_{u=1}^{u=4}\left({\left[-\ln (1-v^2)\right]}_0^{1/2}-{\left[v^2\right]}_0^{1/2}\right) \\ \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\left(\frac{1}{4}\ln \left(\frac{4}{3}\right)-\frac{1}{8}\right){\int }_{u=1}^{u=4}du \\ \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=3\times \left(\frac{1}{4}\ln \left(\frac{4}{3}\right)-\frac{1}{8}\right) \\ \int {\int }_{\Omega }{x}^2\left(1-\frac{1}{y^2}\right) dA=\left(\frac{3}{4}\ln \left(\frac{4}{3}\right)-\frac{3}{8}\right) \end{gathered}$$