13.5 Triple Integrals in Cylindrical and Spherical Coordinates Problem 11 **Transcription for Educational Website:****Task 11:**Consider the set \((r, \theta, z)\) where the following conditions are satisfied:- \(0 \leq r \leq 3\)- \(0 \leq \theta \leq \pi/3\)- \(1 \leq z \leq 4\)These inequalities describe a region in cylindrical coordinates. In this space, \(r\) represents the radial distance, \(\theta\) is the angular component (in radians), and \(z\) is the height. The defined region thus forms a subsection of a cylindrical wedge with a specific height range. **Basic Skills****11–14. Sets in cylindrical coordinates** Identify and sketch the following sets in cylindrical coordinates.

Given set {(r,θ,z):0⩽r⩽3,0⩽θ⩽π/3, 1⩽z⩽4} is in spherical coordinates. We Know that in cylindrical coordinate set will be like {(x,y,z): x=rcosθ,y=rsinθ, r=x2+y2 ,z=z} So, we have to find the values of x and y as z will remain same as in the spherical coordinate set. Now, if we see the region in two dimension, we will see the region be like for θ=π/3 we have equation of line y= tan(π/3 )x i.e. y= 3x and for r=3 , we have x2+y2=9 Now, we will find point of intersection of line y=3x and circle x2+y2=9 we have, x2+(3x)2=9 ⇒x2+3x2=9⇒4x2=9⇒x=±32 our suitable x=3/2, so for x=3/2 ,y=332 So, (32,332) is the point of intersection. So x varies from y3 to 9-y2 and y varies from 0 to 332 So, the given set in cylindrical coordinates is {(x,y,z): y3≤x≤9-y2,0≤y≤332,1≤z≤4}