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