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Set up the integral that gives the volume of the solid E bounded by x = y 2 + z 2 and x = a 2 , where a > 0 .

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Find the average value of the function f ( x , y , z ) = x + y + z over the parallelepiped determined by x = 0 , x = 1 , y = 0 , y = 3 , z = 0 , and z = 5 .

9 2

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Find the average value of the function f ( x , y , z ) = x y z over the solid E = [ 0 , 1 ] × [ 0 , 1 ] × [ 0 , 1 ] situated in the first octant.

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Find the volume of the solid E that lies under the plane x + y + z = 9 and whose projection onto the x y -plane is bounded by x = y 1 , x = 0 , and x + y = 7 .

156 5

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Find the volume of the solid E that lies under the plane 2 x + y + z = 8 and whose projection onto the x y -plane is bounded by x = sin −1 y , y = 0 , and x = π 2 .

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Consider the pyramid with the base in the x y -plane of [ −2 , 2 ] × [ −2 , 2 ] and the vertex at the point ( 0 , 0 , 8 ) .

  1. Show that the equations of the planes of the lateral faces of the pyramid are 4 y + z = 8 , 4 y z = −8 , 4 x + z = 8 , and −4 x + z = 8 .
  2. Find the volume of the pyramid.

a. Answers may vary; b. 128 3

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Consider the pyramid with the base in the x y -plane of [ −3 , 3 ] × [ −3 , 3 ] and the vertex at the point ( 0 , 0 , 9 ) .

  1. Show that the equations of the planes of the side faces of the pyramid are 3 y + z = 9 , 3 y + z = 9 , y = 0 and x = 0 .
  2. Find the volume of the pyramid.
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The solid E bounded by the sphere of equation x 2 + y 2 + z 2 = r 2 with r > 0 and located in the first octant is represented in the following figure.

The eighth of a sphere of radius 2 with center at the origin for positive x, y, and z.
  1. Write the triple integral that gives the volume of E by integrating first with respect to z , then with y , and then with x .
  2. Rewrite the integral in part a. as an equivalent integral in five other orders.

a. 0 4 0 r 2 x 2 0 r 2 x 2 y 2 d z d y d x ; b. 0 2 0 r 2 y 2 0 r 2 x 2 y 2 d z d x d y , 0 r 0 r 2 z 2 0 r 2 x 2 z 2 d y d x d z , 0 r 0 r 2 x 2 0 r 2 x 2 z 2 d y d z d x , 0 r 0 r 2 z 2 0 r 2 y 2 z 2 d x d y d z , 0 r 0 r 2 y 2 0 r 2 y 2 z 2 d x d z d y

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The solid E bounded by the sphere of equation 9 x 2 + 4 y 2 + z 2 = 1 and located in the first octant is represented in the following figure.

In the first octant, a complex shape is shown that is roughly a solid ovoid with center the origin, height 1, width 0.5, and length 0.35.
  1. Write the triple integral that gives the volume of E by integrating first with respect to z , then with y , and then with x .
  2. Rewrite the integral in part a. as an equivalent integral in five other orders.
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Find the volume of the prism with vertices ( 0 , 0 , 0 ) , ( 2 , 0 , 0 ) , ( 2 , 3 , 0 ) , ( 0 , 3 , 0 ) , ( 0 , 0 , 1 ) , and ( 2 , 0 , 1 ) .

3

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Find the volume of the prism with vertices ( 0 , 0 , 0 ) , ( 4 , 0 , 0 ) , ( 4 , 6 , 0 ) , ( 0 , 6 , 0 ) , ( 0 , 0 , 1 ) , and ( 4 , 0 , 1 ) .

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The solid E bounded by z = 10 2 x y and situated in the first octant is given in the following figure. Find the volume of the solid.

A tetrahedron bounded by the x y, y z, and x z planes and a triangle with vertices (0, 0, 10), (5, 0, 0), and (0, 10, 0).

250 3

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The solid E bounded by z = 1 x 2 and situated in the first octant is given in the following figure. Find the volume of the solid.

A complex shape in the first octant with height 1, width 5, and length 1. The shape appears to be a slightly deformed quarter of a cylinder of radius 1 and width 5.
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The midpoint rule for the triple integral B f ( x , y , z ) d V over the rectangular solid box B is a generalization of the midpoint rule for double integrals. The region B is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum i = 1 l j = 1 m k = 1 n f ( x i , y j , z k ) Δ V , where ( x i , y j , z k ) is the center of the box B i j k and Δ V is the volume of each subbox. Apply the midpoint rule to approximate B x 2 d V over the solid B = { ( x , y , z ) | 0 x 1 , 0 y 1 , 0 z 1 } by using a partition of eight cubes of equal size. Round your answer to three decimal places.

5 16 0.313

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[T]

  1. Apply the midpoint rule to approximate B e x 2 d V over the solid B = { ( x , y , z ) | 0 x 1 , 0 y 1 , 0 z 1 } by using a partition of eight cubes of equal size. Round your answer to three decimal places.
  2. Use a CAS to improve the above integral approximation in the case of a partition of n 3 cubes of equal size, where n = 3 , 4 ,…, 10 .
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Suppose that the temperature in degrees Celsius at a point ( x , y , z ) of a solid E bounded by the coordinate planes and x + y + z = 5 is T ( x , y , z ) = x z + 5 z + 10 . Find the average temperature over the solid.

35 2

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Suppose that the temperature in degrees Fahrenheit at a point ( x , y , z ) of a solid E bounded by the coordinate planes and x + y + z = 5 is T ( x , y , z ) = x + y + x y . Find the average temperature over the solid.

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Show that the volume of a right square pyramid of height h and side length a is v = h a 2 3 by using triple integrals.

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Show that the volume of a regular right hexagonal prism of edge length a is 3 a 3 3 2 by using triple integrals.

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Show that the volume of a regular right hexagonal pyramid of edge length a is a 3 3 2 by using triple integrals.

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If the charge density at an arbitrary point ( x , y , z ) of a solid E is given by the function ρ ( x , y , z ) , then the total charge inside the solid is defined as the triple integral E ρ ( x , y , z ) d V . Assume that the charge density of the solid E enclosed by the paraboloids x = 5 y 2 z 2 and x = y 2 + z 2 5 is equal to the distance from an arbitrary point of E to the origin. Set up the integral that gives the total charge inside the solid E .

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Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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