Question

Standard 24: (a) Write the integral $\iiint_E \sqrt{x^2 + y^2} dV$ in cylindrical coordinates, where $E$ is the region that lies inside the cylinder $x^2 + y^2 = 16$ and between the planes $z = -5$ and $z = 4$. You do not need to evaluate the integral. (b) Write the integral $\iiint_H (9 - x^2 - y^2) dV$ in spherical coordinates, where $H$ is the solid hemisphere $x^2 + y^2 + z^2 \le 9$, $z \ge 0$. You do not need to evaluate the integral.

          Standard 24: (a) Write the integral $\iiint_E \sqrt{x^2 + y^2} dV$ in cylindrical coordinates, where $E$ is the region that lies inside the cylinder $x^2 + y^2 = 16$ and between the planes $z = -5$ and $z = 4$. You do not need to evaluate the integral.
(b) Write the integral $\iiint_H (9 - x^2 - y^2) dV$ in spherical coordinates, where $H$ is the solid hemisphere $x^2 + y^2 + z^2 \le 9$, $z \ge 0$. You do not need to evaluate the integral.

Standard 24: (a) Write the integral √(x^2 + y^2) dV in cylindrical coordinates, where E is the region that lies inside the cylinder x^2 + y^2 = 16 and between the planes z = -5 and z = 4. You do not need to evaluate the integral.
(b) Write the integral (9 - x^2 - y^2) dV in spherical coordinates, where H is the solid hemisphere x^2 + y^2 + z^2 ≤ 9, z ≥ 0. You do not need to evaluate the integral.

Added by Julie C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

08/21/2023

Step 1

The region E lies inside the cylinder x^(2)+y^(2)=16 and between the planes z=-5 and z=4. In cylindrical coordinates, the cylinder x^(2)+y^(2)=16 becomes r=4. The region E can be expressed as 4≥r≥0, -5≤z≤4, and 0≤θ≤2π. Show more…

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Standard 24: (a) Write the integral ∭_(E)√(x^(2)+y^(2))dV in cylindrical coordinates, where E is the region that lies inside the cylinder x^(2)+y^(2)=16 and between the planes z=-5 and z=4. You do not need to evaluate the integral. (b) Write the integral ∭_(H)(9-x^(2)-y^(2))dV in spherical coordinates, where H is the solid hemisphere x^(2)+y^(2)+z^(2)≤9, z≥0. You do not need to evaluate the integral.