VolumeBetweenSurfaceCylinder::usage="Use polar coordinates to calculate the volume of the solid that is below the surface z = (​
2
x
+ 1)y and above the disc surrounded by the circumference
2
x
+ (y - 1
2
)
= 1. References: Hollis, Selwyn. CalcLabs with Mathematica, Multivariable Calculus. 4a. Ed.: p.121, 2010; Personal.";
​
Development:
In[]:=
eqn:=
2
x
+
2
(y-1)
==1​​eqn/.{xrCos[θ],yrSin[θ]}//Simplify​​
Out[]=
2
r
2rSin[θ]
In[]:=
​​Manipulatef[x_,y_]:=(
2
x
+1)y;​​V=
π
∫
0
2Sin[θ]
∫
0
f[rCos[θ],rSin[θ]]rrθ;​​p=Grid[{{Volume},{V}},FrameAll,ItemSize->15,ItemStyle->15];​​Plot3D[f[x,y],{x,-1.1,1.1},{y,-0.1,t},BoxRatiosAutomatic,RegionFunctionFunction[{x,y},
2
x
+
2
(y-1)
<=1],FillingBottom,MaxRecursion->3],{t,-0.09,2.1,0.01},Delimiter,{{p,2,"Volume"}},Delimiter,SaveDefinitions->True,ControlPlacement->Top
Out[]=
​
t
Volume
Volume
5π
4
Another method (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion[z<=(
2
x
+1)y,{x,y,z}];​​s2=ImplicitRegion[
2
x
+
2
(y-1)
<=1,{x,y,z}];​​s3=ImplicitRegion[z>=0,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
5π
4