rl-у с1-г2 1. Write 5 other integrals that equal to the iterated integral /|| h(x, y, z)dzdzdy. /0 2. Suppose b > a > 0. Consider a sphere of radius b, and a circular cylinder of radius a, both centered at the origin. Consider the solid ring formed by all the points which are both inside the sphere and outside the cylinder. Imagine boring a hole with a cylindrical drill, and punching it all the way through. (a) Find the height h of the ring. You should express h in terms of a and b. Please accompany with a labeled sketch. (b) Find the volume of the ring, in terms of a and b. (c) Express the volume in terms of the height h of the ring only. 3. You wish to find the mass of a ball described by x + y? + 2? < c?. The density of the ball at any point P(r, y, 2) is proportional to the distance from the point to the :-axis. (a) Set us the integral in rectangular coordinates of the mass of this ball. Do not integrate. (b) Set up the integral from (a) in terms of cylindrical coordinates. Do not integrate. (c) Set up the integral from (a) in terms of Spherical coordinates. Do not integrate. (d) Now, find the mass of the ball. Here, you may want to use the setup from (a)-(c) that is easiest to integrate. 6- 12 - y? and 4. You wish to find the volume of a solid that lies between the paraboloid z = the cone z = V3.r2 + 3y?. (a) Set us the volume as a double integral. Do not integrate. 1 (b) Set us the volume as a triple integral using i. Rectangular coordinates. ii. Cylindrical coordinates. iii. Spherical coordinates. iv. Find the Volume of the solid. Again, you may want to use the setup from (a) or (b)i, ii, iii that is easiest to integrate.
rl-у с1-г2 1. Write 5 other integrals that equal to the iterated integral /|| h(x, y, z)dzdzdy. /0 2. Suppose b > a > 0. Consider a sphere of radius b, and a circular cylinder of radius a, both centered at the origin. Consider the solid ring formed by all the points which are both inside the sphere and outside the cylinder. Imagine boring a hole with a cylindrical drill, and punching it all the way through. (a) Find the height h of the ring. You should express h in terms of a and b. Please accompany with a labeled sketch. (b) Find the volume of the ring, in terms of a and b. (c) Express the volume in terms of the height h of the ring only. 3. You wish to find the mass of a ball described by x + y? + 2? < c?. The density of the ball at any point P(r, y, 2) is proportional to the distance from the point to the :-axis. (a) Set us the integral in rectangular coordinates of the mass of this ball. Do not integrate. (b) Set up the integral from (a) in terms of cylindrical coordinates. Do not integrate. (c) Set up the integral from (a) in terms of Spherical coordinates. Do not integrate. (d) Now, find the mass of the ball. Here, you may want to use the setup from (a)-(c) that is easiest to integrate. 6- 12 - y? and 4. You wish to find the volume of a solid that lies between the paraboloid z = the cone z = V3.r2 + 3y?. (a) Set us the volume as a double integral. Do not integrate. 1 (b) Set us the volume as a triple integral using i. Rectangular coordinates. ii. Cylindrical coordinates. iii. Spherical coordinates. iv. Find the Volume of the solid. Again, you may want to use the setup from (a) or (b)i, ii, iii that is easiest to integrate.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:rl-у с1-г2
1. Write 5 other integrals that equal to the iterated integral /||
h(x, y, z)dzdzdy.
/0
2. Suppose b > a > 0. Consider a sphere of radius b, and a circular cylinder of radius a,
both centered at the origin. Consider the solid ring formed by all the points which are both
inside the sphere and outside the cylinder. Imagine boring a hole with a cylindrical drill, and
punching it all the way through.
(a) Find the height h of the ring. You should express h in terms of a and b. Please accompany
with a labeled sketch.
(b) Find the volume of the ring, in terms of a and b.
(c) Express the volume in terms of the height h of the ring only.
3. You wish to find the mass of a ball described by x + y? + 2? < c?. The density of the ball
at any point P(r, y, 2) is proportional to the distance from the point to the :-axis.
(a) Set us the integral in rectangular coordinates of the mass of this ball. Do not integrate.
(b) Set up the integral from (a) in terms of cylindrical coordinates. Do not integrate.
(c) Set up the integral from (a) in terms of Spherical coordinates. Do not integrate.
(d) Now, find the mass of the ball. Here, you may want to use the setup from (a)-(c) that
is easiest to integrate.
6- 12 - y? and
4. You wish to find the volume of a solid that lies between the paraboloid z =
the cone z =
V3.r2 + 3y?.
(a) Set us the volume as a double integral. Do not integrate.
1
(b) Set us the volume as a triple integral using
i. Rectangular coordinates.
ii. Cylindrical coordinates.
iii. Spherical coordinates.
iv. Find the Volume of the solid. Again, you may want to use the setup from (a) or
(b)i, ii, iii that is easiest to integrate.
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