Converting triple integrals to cylindrical coordinates (KristaKingMath)
Converting triple integrals to cylindrical coordinates (KristaKingMath)

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

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