91影视

To convert the given integral into cylindrical coordinates, we'll replace the Cartesian coordinates (x, y, z) with cylindrical coordinates (蟻, 蠁, z) using the following transformations: - \(x = 蟻\cos{蠁}\) - \(y = 蟻\sin{蠁}\) - \(z = z\) Additionally, the Jacobian for the transformation between Cartesian and cylindrical coordinates is \(蟻\). So, the integral becomes: $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{x^{2}+y^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} \rho d z d y d x$$ Now, let's plug in the cylindrical coordinates conversions and change the integral limits.

After converting to cylindrical coordinates, we rewrite the limits of integration: - For \(x\), we know that \(x = \rho\cos{蠁}\), and the integral limits are from \(0\) to \(3\). So, \(\rho\cos{蠁}\) ranges from \(0\) to \(3\). Therefore, we have \(蟻\) ranges from \(0\) to \(3\). - For \(y\), we know that \(y = \rho\sin{蠁}\), and the integral limits are from \(0\) to \(\sqrt{9-x^{2}}\). So, \(\rho\sin{蠁}\) ranges from \(0\) to \(\sqrt{9-(\rho\cos{蠁})^{2}}\). This implies that \(蠁\) ranges from \(0\) to \(\pi/2\). - For \(z\), the integral limits are from \(0\) to \(\sqrt{x^{2}+y^{2}} = \rho\). With the new limits and the Jacobian, the integral becomes: $$\int_{0}^{3} \int_{0}^{\pi/2} \int_{0}^{\rho}\rho^{-1} \rho d z d\phi d\rho$$

Simplify the integral: $$\int_{0}^{3} \int_{0}^{\pi/2} \int_{0}^{\rho} d z d\phi d\rho$$ Now, we can solve the integral for \(z\), \(蠁\), and \(蟻\) one by one.

First, solve the inner integral with respect to \(z\): $$\int_{0}^{3} \int_{0}^{\pi/2} \left[z\right]_{0}^{\rho} d\phi d\rho = \int_{0}^{3} \int_{0}^{\pi/2} \rho d\phi d\rho$$

Next, solve the middle integral with respect to \(蠁\): $$\int_{0}^{3} \left[\rho\phi\right]_{0}^{\pi/2} d\rho = \int_{0}^{3} \frac{\pi}{2}\rho d\rho$$

Finally, solve the outer integral with respect to \(蟻\): $$\left[\frac{\pi}{4}\rho^2\right]_{0}^{3} = \frac{9\pi}{4} - 0 = \boxed{\frac{9\pi}{4}}$$ The value of the given integral in cylindrical coordinates is \(\frac{9\pi}{4}\).