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--- book:chap5:5.3_volume_integrals [2022/01/04 20:20] – [5.3.3 The force of an extended object (Earth) on a point particle (professor)] abril
+++ book:chap5:5.3_volume_integrals [2022/01/04 20:29] (current) – abril
@@ Line 1: / Line 1: @@
-FIXME draft with missing figures and references :!:
 ===== 5.3  Volume integrals — A professor falling through Earth  =====
 <WRAP #section_VolumeMassDensity></WRAP>
@@ Line 127: / Line 125: @@
-<WRAP box round>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\
+<WRAP box round #bsp_VolumeIntegral>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\
 The volume of a three-dimensional sphere $S$ with center at the origin and radius $R$ is
 \begin{align*}
@@ Line 260: / Line 258: @@
 </WRAP>
-<WRAP box round>**Example 5.3** <wrap em>Integration volumes</wrap> \\
+<WRAP box round #bsp_IntegrationVolumes>**Example 5.3** <wrap em>Integration volumes</wrap> \\
 **a)** //polar coordinates//
@@ Line 375: / Line 373: @@
 [[book:chap4:4.10_problems #quest_Conservation-08 |Problem 4.30]]!
-After a while (cf [[#quest_volIntegral-professor |Problem 5.12]]) he reappears at the very same spot where he started,
+After a while (cf [[book:chap5:5.7_problems #quest_volIntegral-professor |Problem 5.12]]) he reappears at the very same spot where he started,
 except that Earth moved on while he was under way.
 ==== 5.3.4 Self Test ====
-----
 <wrap #quest_volIntegral-parallelogram > Problem 5.5: </wrap>** Area of a parallelogram **
@@ Line 400: / Line 396: @@
 describes a sphere of radius  $R$.
 The volume  $V$ of a solid of revolution are given by the integral
-\begin{align}\label{eq:revolutionSolidVolume}
-    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2
+<wrap #eq_revolutionSolidVolume></wrap>
+\begin{align}
+    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2 \tag{5.3.3}
 \end{align}
-  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$
+  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$ and verify that the solid of revolution is indeed a sphere.
-and verify that the solid of revolution is indeed a sphere.
+  -  Determine the volume of the sphere based on the given equation Compare you calculation and the result to the calculation given in [[#bsp_VolumeIntegral |Example 5.2]].
-  -  Determine the volume of the sphere based on the given equation.
+  -  Show that the volume integral for a solid of revolution provides [[#eq_revolutionSolidVolume |Equation 5.3.3]]  when one adopts cylindrical coordinates.
-Compare you calculation and the result to the calculation given in \Example{VolumeIntegral}.
-  -  Show that the volume integral for a solid of revolution provides
-\cref{eq:revolutionSolidVolume}
-when one adopts cylindrical coordinates.
@@ Line 430: / Line 424: @@
 \\
 Determine the Jacobi matrix and its determinant for the transformation
-from Cartesian to spherical coordinates, \cf\Example{IntegrationVolumes}c).
+from Cartesian to spherical coordinates, cf [[#bsp_IntegrationVolumes |Example 5.3 c)]].
 ~~DISCUSSION~~