Question

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ x y=9, y=x, y=0, x=9 $$

Short answer

The area of the region bounded by the given curves is \( 9\ln9 \) square units.

Step-by-step solution

Interpret the Given Equations

First, interpret the given equations. \( xy=9 \) represents a hyperbola, \( y=x \) is a straight line through the origin, \( y=0 \) is the x-axis, and \( x=9 \) represents a vertical line at \( x=9 \).

Set Up the Double Integral

To find the area, set up a double integral of the function \( f(x, y) = 1 \) over the region bounded by the curves. The limits of \( x \) will be from \( 0 \) to \( 9 \), and the limits of \( y \) will be from \( 0 \) to \( \frac{9}{x} \). Thus, the double integral is \[ \int_{0}^{9} \int_{0}^{\frac{9}{x}} dy \, dx \].

Evaluate the Inner Integral

Next, evaluate the inner integral with respect to \( y \), which just gives \( \frac{9}{x} \).

Evaluate the Outer Integral

Finally, evaluate the outer integral with respect to \( x \), which results in the area. This is \[ \int_{0}^{9}\frac{9}{x} dx = 9\ln|x| \Big|_0^9 = 9(\ln|9|- \ln|0|)=9\ln9 \].