7/8/13
Vol umes of Sol i ds of R evolution | Ap Appli cati ons of Integ r ati on | In Integ r al Calcul us Review
Class 1 to Class 12 Lessons, Animations, Videos & more… Math, EVS, Science, English, SST… www.MeritNation.com/CBSE Colgate Total® Toothpaste Helps you Reverse Reverse Gum Problems in just 4 Weeks. Weeks. Click to know more www.colgate.co.in/colgatetota activities, less on plans & worksheets. www.TESIndia.com Free Teacher W orksheets Thousands of free class activities,
IIT JEE Maths Material Online Video Lectures. India's #1 Video Coaching. Join Now ExponentEducation.com/JEE_Maths
Volumes of Solids of Revolution | Applications of Integration Tags: vol volum ume e sol solid ids s of revoluti u tion o n vol volum ume e by integrat integratio ion n cyli cylind ndrica ricall shell meth method od circu circula larr disk meth method od
Solids of Rev R evoluti olution on by Integration The solid generated by rotating a plane area about an axis in its plane is called a solid solid of rev olution olution . The volume of a solid of revolution may be found by the following procedures:
Circular Disk Method The strip that will revolve is perpendicular to the axis of revolution. In this method, the axis of rotation may or may not be part of the boundary of the plane area that is being revolved. Using Horizontal Strip
The disk as shown in the figure has an outer radius of xR, a hole of radius xL, and thickness dy. The differential volume is therefore π xR2 dy -
π
xL2 dy
and the total volume is...
The integration involved is in variable y since the derivative is dy, xR and xL therefore must be expressed in terms of y. If the axis of revolution is part of the boundary of the plane area that is being revolved, xL = 0, and the equation reduces to...
Using Vertical Strip
From the figure shown below, the volume can be found by the formula...
www.mathal ino.com/r eviewer /i nteg r al - calcul us/vol umes- of- sol i ds- of- r evol uti on- appl i cati ons- of- i nteg r ation
1/4
7/8/13
Volumes of Solids of Revolution | Applications of Integration | Integral Calculus Review
If yL = 0, we have
$
Where yU and yL are functions of x.
Cylindrical Shell Method The strip that will revolve is parallel to the axis of revolution. The volume of revolution is obtained by taking the limit of the sum of cylindrical shell elements, each of which is equal in volume to the mean circumference times the height times the thickness. Using Horizontal Strip
Using Vertical Strip
www.mathalino.com/reviewer/integral-calculus/volumes-of-solids-of-revolution-applications-of-integration
2/4
VAV Boxes www.tritonprocess.com UL/CE/AHRI Ce rtified Bacnet VAV Box a t Unbelievable prices
Class 1 to C lass 12 www.MeritNation.com/CBSE Lessons, Animations, Videos & more… Math, EVS, Science, English, SST…
Colgate Total® Toothpaste www.colgate.co.in/colgatetotal* Helps you Reverse Gum Problems in just 4 Weeks. C lick to know more
Free Teacher Wo rksheets www.TESIndia.com Thousands of free class activities, lesson plans & worksheets.
► Integration ► Integral Calculus ► Math Help Calculus
