Describe the Solid Using Spherical Coordinates

We can describe a point P in three different ways. R n ρφθ.

Triple Integrals In Cylindrical And Spherical Coordinates Calculus Volume 3

The other way to get this range is from the cone by itself.

. Solution for Use spherical coordinates to describe the region above the xy-plane between the spheres of radius 1 and 3 centered at the origin. X ρ sin φ cos θ 8 sin π 6 cos π 3 8 1 2 1 2 2 y ρ sin φ sin θ 8 sin π 6 sin π 3 8 1 2 3 2 2 3 z ρ cos φ 8 cos π 6 8 3 2 4 3. The cone is z 3 x 2 y 2.

A Describe the solid using spherical coordinates. Position the solid on the coordinate system such that the inequalities are as simple as possible. A portion of the z sphere of radius 3.

Beginning with the line θ π 3 in the x-y plane as given by polar coordinates extend the line parallel to the z-axis forming a plane. Describe the solid in terms of inequalities involving spherical coordinates. This means that the iterated integral is Z 2ˇ 0 Z ˇ4 0 Z 1 0 ˆcosˆ2 sindˆdd.

For the remaining problems use the coordinate system Cartesian. Find inequalities that describe the solid and state the coordinate system used. Your final answer must be written in set-builder notation.

Use the formulas in 8 to express i j k in terms of er ee and ez. So the solid can be described in spherical coordinates as 0 ˆ 1 0 ˇ 4 0 2ˇ. θ 02π φ h 0 π 4 i ρ 01 o.

Describe the solid in spherical coordinates. I graphed the solid and it looks like this. 0 φ π 4 0 φ π 4.

Position the solid on the coordinate system such that the inequalities are as simple as possible. X ρ cos. Triple integral in spherical coordinates Example Use spherical coordinates to find the volume below the sphere x2 y2 z2 1 and above the cone z p x2 y2.

Describe using spherical coordinates the solid E in the first octant that lies above the half-cone zx 2 y 2 but inside x 2 y 2 z 2 1. A portion of the sphere of radius 3. In the cylindrical coordinate system location of a point in space is described using two distances and an angle measure In the spherical coordinate system we again use an ordered triple to describe the location of a point in space.

Consider the solid in R3 shown in Figure 1110. Homework Equations In spherical coordinates texxrhosinphicosthetatex texyrhosinphisinthetatex and texzrhocosphitex The Attempt at a Solution. Homework Equations ρ x 2 y 2 z 2 x ρsinφcosθ y ρsinφsinθ z ρcosφ The Attempt at a Solution.

The first thing that well do here is find ρ ρ. The solid between the spheres x 2y 2z 24 and x 2y 2z 29 and inside the cone z 2x 2y 2 Problem. Homework Equations In spherical coordinates texxrhosinphicosthetatex texyrhosinphisinthetatex and texzrhocosphitex The Attempt at a Solution.

ˆ 1 in spherical coordinates. SOLVEDFind inequalities that describe the solid and state the coordinate system used. Describe the solid using spherical coordinates.

Also for each θ ϕ the value of ρ ranges from 0 until the surface of the cone. The equation θ π 3 describes the same surface in spherical coordinates as it does in cylindrical coordinates. V Z 2π 0 Z π4 0 Z 1 0 ρ2.

Describe the solid in terms of inequalities involving spherical coordinates. So clearly 0 θ π and 0 ϕ π 2. Use the equations in Converting among Spherical Cylindrical and Rectangular Coordinates to translate between spherical and cylindrical coordinates Figure 2100.

C Consider figure a shown below. A solid lies above the cone texzsqrtx2z2tex and below the sphere texx2y2z2ztex. So the spherical coordinates of this point will are 2 2 π 4 π 3 2 2 π 4 π 3.

ρ x 2. In this case the. The solid that remains after a hole 1 inch in diameter is drilled through the center of a sphere 6 inches in diameter The solid inside both x2y2z29 and leftx.

B Describe the solid using cylindrical coordinates. X ρ sin φ cos θ 8 sin π 6 cos π 3 8 1 2 1 2. Use the formulas in 9 to express i j k in terms of ep eq and ee.

B Convert the point 112 1 1 2 from Cartesian to spherical coordinates. By first converting the equation into cylindrical coordinates and then into spherical coordinates we get the following z r ρ cos φ ρ sin φ 1 tan φ φ π 4 z r ρ cos φ ρ sin φ 1 tan. Describe the solid using cylindrical coordinates.

The calculation is simple the region is a simple section of a sphere. A solid lies above the cone texzsqrtx2z2tex and below the sphere texx2y2z2ztex. Cartesian Cylindrical Spherical Cylindrical Coordinates x r cosθ r x2 y2 y r sinθ tan θ yx z z z z Spherical Coordinates x ρsinφcosθ ρ x2 y2 z2 y ρsinφsinθ tan θ yx z ρcosφ cosφ x2 y2 z2 z.

Solved The Solid Bounded Below By The Sphere Rho 2 Cos Phi And Above By The Cone Z Sqrt X 2 Y 2

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