Question

The integral,where \(R = \left( {0,4} \right) \times \left( {0,2} \right)\)represents the volume of a solid. Sketch the solid.

Short answer

Given integral is,\(R = \left( {0,4} \right) \times \left( {0,2} \right)\)represents the volume of a solid. Sketch the solid.

Step-by-step solution

Given data:

Consider the integral, where\(R = \left( {0,4} \right) \times \left( {0,2} \right)\)

To sketch the solid, use the given information

Let\(f\left( {x,y} \right) = \sqrt {9 - {y^2}} \)

The function\(f\left( {x,y} \right) = \sqrt {9 - {y^2}} \)is always positive when\(0 \le y \le 2\) .

If\(z = \sqrt {9 - {y^2}} \)then \({y^2} + {z^2} = 9\)and\(z \ge 0\).

So the given double integral represents the volume of the solid that lies below the circular cylinder \({y^2} + {z^2} = 9\) and above the rectangle R.

Step 2:Plotting graph:-

The graph of the solid is shown below.

Hence, the given double integral represents the volume of the solid that lies below the circular cylinder \({y^2} + {z^2} = 9\) and above the rectangle R.