# Double Integrals: Volume of a Solid in the First Octant

#### pandalover305

How would you find the volume in the first octant of the solid bounded by the cylinder y2+z2=9 and x=6?

I tried doing a double integral, both from 0 to 3 of (y2+z2)dydz then multiply by 6, but I keep getting 0.

#### HallsofIvy

y and z both going from 0 to 3 gives a rectangle.

$y^2+ z^2= 9$ is a circle of radius 3. What is the area of that circle? What is the area of 1/4 of that circle? Since x= 6 is parallel to the yz-plane, just multiply the area of 1/4 the circle by 6. No integration required.

(If you really want to do an integration, then you could take y from 0 to 3 but then z, for each y, must be from 0 to $z= \sqrt{9- y^2}$. That is, the area is
$$\int_{y=0}^3\int_{z= 0}^{\sqrt{9- y^2}}dzdy\right)$$

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