Using cylindrical coordinates evaluate the integral \[\int\limits_{ - 2}^2 {dx} \int\limits_{ - \sqrt {4 - {x^2}} }^{\sqrt {4 - {x^2}} } {dy} \int\limits_0^{4 - {x^2} - {y^2}} {{y^2}dz} .\]
Example 4
Calculate the integral using cylindrical coordinates: \[\iiint\limits_U {\sqrt {{x^2} + {y^2}} dxdydz} .\] The region U is bounded by the paraboloid z = 4 − x² − y², by the cylinder x² + y² = 4 and by the planes y = 0, z = 0 (Figure 8).
Example 5
Find the integral \[\iiint\limits_U {ydxdydz},\] where the region \(U\) is bounded by the planes \(z = x + 1,\) \(z = 0\) and by the cylindrical surfaces \({x^2} + {y^2} = 1,\) \({x^2} + {y^2} = 4\) (see Figure \(10\)).
Example 3.
Using cylindrical coordinates evaluate the integral \[\int\limits_{ - 2}^2 {dx} \int\limits_{ - \sqrt {4 - {x^2}} }^{\sqrt {4 - {x^2}} } {dy} \int\limits_0^{4 - {x^2} - {y^2}} {{y^2}dz} .\]
Solution.
The region of integration \(U\) is shown in Figure \(6.\) Its projection on the \(xy\)-plane is the circle \({x^2} + {y^2} = {2^2}\) (Figure \(7\)).
Figure 6.Figure 7.
The new variables in the cylindrical coordinates range within the limits:
Calculate the integral using cylindrical coordinates: \[\iiint\limits_U {\sqrt {{x^2} + {y^2}} dxdydz} .\] The region \(U\) is bounded by the paraboloid \(z = 4 - {x^2} - {y^2},\) by the cylinder \({x^2} + {y^2} = 4\) and by the planes \(y = 0,\) \(z = 0\) (Figure \(8\text{).}\)
Solution.
By sketching the region of integration \(U\) (Figure \(9\)), we see that its projection on the \(xy\)-plane (the region \(D\)) is the half-circle of radius \(\rho = 2.\)
Figure 8.Figure 9.
We convert to cylindrical coordinates using the substitutions
Find the integral \[\iiint\limits_U {ydxdydz},\] where the region \(U\) is bounded by the planes \(z = x + 1,\) \(z = 0\) and by the cylindrical surfaces \({x^2} + {y^2} = 1,\) \({x^2} + {y^2} = 4\) (see Figure \(10\)).
Solution.
We calculate this integral in cylindrical coordinates. From the condition
\[0 \le z \le x + 1\]
it follows that
\[0 \le z \le \rho \cos \varphi + 1.\]
The projection of the region of integration onto the \(xy\)-plane is the ring formed by the two circles: \({x^2} + {y^2} = 1\) and \({x^2} + {y^2} = 4\) (Figure \(11\)). Hence, the variables \(\rho\) and \(\varphi\) range in the interval