Question

Set up a double integral to find the volume of the solid bounded by the graphs of the equations. \(z=x+y, x^{2}+y^{2}=4,\) first octant

Short answer

The volume of the solid bounded by the given equations in the first octant is \( \frac{16}{3} \) cubic units.

Step-by-step solution

Analyze the equations

The equation \(x^{2}+y^{2}=4\) represents a cylinder with a radius of 2. The equation \(z=x+y\) represents a plane that intersects the positive z-axis at the point (0,0,4). Because we're considering the first octant, all x, y, and z values will be positive.

Convert to polar coordinates

To proceed, we'll convert the cylinder's equation into polar coordinates. In the polar coordinate system, \(x^{2}+y^{2}\) becomes \(r^{2}\). Therefore, the radius r will vary between 0 and 2. The angle \(\theta\) will vary between 0 and \(\pi/2\) to satisfy the first octant condition.

Set up the double integral

The volume V of the solid bounded by the graph can be found by integrating the function representing the upper boundary over the area. So, we'll setup the double integral for the function \(f(x, y) = x+y\) in polar coordinates as \(z=r cos(\theta) + r sin(\theta)\) which will give us our double integral as \( V = \int_0^{\pi/2} \int_0^{2} r [r cos(\theta) + r sin(\theta)] dr d\theta \).

Evaluate the Integral

After evaluating this double integral, the first gives us \( \frac{1}{2} r^{3} cos(\theta) + \frac{1}{2} r^{3} sin(\theta) |_0^{2} \). Calculating the outer integral, we obtain \( V = \frac{8}{3} [sin(\theta)- cos(\theta)] |_0^{\pi/2} \). This yields a final volume V = \( \frac{16}{3} \) cubic units.