[PDF] Homework 24: Cylindrical/Spherical integration PDF hw24.pdf

)dV , where H is the solid hemisphere x 2 + y 2 + z 2 ≤ 16, z ≥ 0 4 Evaluate the integral by changing to spherical coordinates ∫ a −a ∫ √ a2−y2 − √

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15 4 Double Integrals in Polar Coordinates 15 9 Triple Integrals in Spherical Coordinates Evaluate the integral by changing to cylindrical coordinates: ∫ 1

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Triple integral in spherical coordinates Cylindrical coordinates in space Definition The cylindrical coordinates of a point P ∈ R3 is the ordered triple (r, θ, z)

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Evaluate the integral using spherical coordinates: dxdydz T ∫∫∫ 16 10 The Jacobian; Changing Variables in Multiple Integration So far, we have used

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Thus to evaluate an integral in spherical coordinates, we do the follow- ing: (i) Convert the function f(x, y, z) into a spherical function (ii) Change the limits of the

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Some regions in space are easier to express in terms of cylindrical or spherical coordinates Triple integrals over these regions are easier to evaluate by

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A Review of Double Integrals in Polar Coordinates where we write ∆r = b−a and ∆θ = d−c (the change in radius and the change (3) Evaluate the integral

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ated integral in polar coordinates to describe this disk: the disk is 0 ≤ r ≤ 2, 0 ≤ θ < 2π, so our iterated To compute this, we need to convert the triple integral an iterated integral which gives the volume of U (You need not evaluate )

[PDF] Homework 24: Cylindrical/Spherical integration

)dV , where H is the solid hemisphere x 2 + y 2 + z 2 ≤ 16, z ≥ 0 4 Evaluate the integral by changing to spherical coordinates ∫ a −a ∫ √ a2−y2 − √

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Unmarked “Homework 10” Solutions 2016 April 8 1 Evaluate ∫ 3 -3 ∫ √ by changing to spherical coordinates Solution: Therefore the given integral is

[PDF] Math 241, Exam 3 11/21/11 Name:

(17 points): Evaluate the integral by changing to spherical coordinates ∫ 4 0 ∫ √ 16−y2 − √ 16−y2 ∫ √ 16−x2−y2 0 (x2 + y2 + z2)z dz dx dy

Math 21a: Multivariable calculus Fall 2019

Homework 24: Cylindrical/Spherical integration

This homework is due Monday, 11/11.

1Evaluate the following integrals.

a) Z 0Z 2 0Z 9r2

07r dz drd

b) Z2 0Z =2Z 2

12sin ddd2Use cylindrical coordinates to nd the volume of the solid that lies

within both the cylinderx2+y2= 1 and the spherex2+y2+z2=

4.3Use spherical coordinates to evaluate

Z Z Z

H(16x2y2)dV ;

whereHis the solid hemispherex2+y2+z216,z0.4Evaluate the integral by changing to spherical coordinates.

Z a aZpa 2y2 pa

2y2Zpa

2x2y2 pa

2x2y2(x2z+y2z+z3)dz dxdy5Show that

Z 1 1Z 1 1Z 1 1rx

2+y2+z2e(x2+y2+z2)dxdy dz= 2 :

Main denitions

The integration factor in cylindrical coordinates

(x;y;z) = (rcos();rsin();z) isras in polar co- ordinates.The integration factor in spherical coordinates (x;y;z) = (sin()cos();sin()sin();cos()) is

2sin(). This was the surface area elementj~r~rjTo evaluate an integral in spherical coordinates, we express the

regioninsphericalcoordinates, substitutex=sin()cos();y= sin()sin() andz=cos()) in the function and include the integration factor2sin(). 2quotesdbs_dbs17.pdfusesText_23

[PDF] [PDF] Homework 24: Cylindrical/Spherical integration