Volumes in spherical coordinates Use integration in spherical coordinates to find the volume of the following solids . 49. The solid rose petal of revolution D = { ( ρ , φ , θ ) : 0 ≤ ρ ≤ 4 sin 2 φ , 0 ≤ φ π / 2 , 0 ≤ θ ≤ 2 π }
Volumes in spherical coordinates Use integration in spherical coordinates to find the volume of the following solids . 49. The solid rose petal of revolution D = { ( ρ , φ , θ ) : 0 ≤ ρ ≤ 4 sin 2 φ , 0 ≤ φ π / 2 , 0 ≤ θ ≤ 2 π }
Solution Summary: The author explains how the volume of the given solid in spherical coordinates is given by, V=displaystyle undersetDiiint
Volumes in spherical coordinatesUse integration in spherical coordinates to find the volume of the following solids.
49. The solid rose petal of revolution
D
=
{
(
ρ
,
φ
,
θ
)
:
0
≤
ρ
≤
4
sin
2
φ
,
0
≤
φ
π
/
2
,
0
≤
θ
≤
2
π
}
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
3. Describe the steps you would take to find the absolute max of the following
function using Calculus f(x) = :
, [-1,2]. Then use a graphing calculator to
x-1
x²-x+1
approximate the absolute max in the closed interval.
(7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz).
Ꮖ
(a) (4 points) Show that V x F = 0.
(b) (4 points) Find a potential f for the vector field F.
(c) (4 points) Let S be a surface in R3 for which the Stokes' Theorem is valid. Use
Stokes' Theorem to calculate the line integral
Jos
F.ds;
as denotes the boundary of S. Explain your answer.
(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY