B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a

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1 Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing Riemnn Sum so tht we cn trnsform them into integrls. B. Definition: The volume of solid of known integrle cross-section re A(x) from x = to x = is the integrl of A from to. V = A( x) dx. C. Steps to Find Volumes y the Method of Slicing 1. Sketch the solid nd typicl cross section.. Find formul for A(x) 3. Find the limits of integrtion. 4. Integrte A(x) to find the volume, i.e., V = A( x) dx. D. Exmples 1. Find the volumes of the solids if the solid lies etween plnes perpendiculr to the x- xis t x = 0 nd x = 4. The cross sections perpendiculr to the x-xis etween these plnes run from the prol y = x to the prol y = x. The cross sections re semi-circulr disks with dimeters in the xy-plne. Note: the re of semicirculr disk is given y A = πr. Find the volume of the solid tht lies etween plnes perpendiculr to the x-xis t x = -1 nd x = 1. In ech cse, the cross sections perpendiculr to the x-xis, etween these plnes, run from the semicircle y = 1 x to the semicircle y = 1 x. Assume the cross sections re squres with ses in the xy-plne.

2 . The cross sections re squres with digonls in the xy-plne. (The length of squre's digonl is times the length of its sides). The se of the solid is the region etween the curve y = sin( x) nd the intervl [0,π] on the x-xis. The cross sections perpendiculr to the x-xis re verticl equilterl tringles with ses running from the x-xis to the curve. Note: the re of tringle is given y A = 1 h or A = 3 4 s 3. The solid lies etween the plnes perpendiculr to the x-xis t x = 3 nd x = 3. The cross sections perpendiculr to the x-xis re circulr disks with dimeters running from the curve y = tn x to the curve y = sec x.

3 II. Volumes of Solids of Revolution Disks & Wshers A. Introduction In this prt of the section, we egin with known plnr shpe nd rotte tht shpe out line. The resulting three dimensionl shpe is known s solid of revolution. The line out which we rotte the plne shpe, is clled the xis of rottion. Certin solids of revolution tht cn e generted like cylinder, cone, or sphere, we cn find their volumes using formuls from geometry. However, when the solid or revolution tkes on non-regulr shpe, like spool, ullet, or limp, etc... there re often times no esy geometric volume formuls. So we fll ck on integrl clculus to compute the volume. B. Disks Method 1. Rottion out the x-xis:. If we revolve continuous function y=r(x), on [,], nd the x-xis, out the x- xis s our xis of revolution, then we hve solid formed (with no hole); the cross section perpendiculr to the xis of revolution will e disk of rdius (R(x)-0) which will hve re A x [ ]. ( ) = ( rdius) = R( x). The Volume of the solid generted y revolving out the x-xis the region etween the x-xis nd the grph of the continuous function y=r(x), <x<, is V = [ R ( x ) ] dx. c. Steps to Find the Volume of the Solid of Revolution 1. Grph the region. Determine the xis of revolution 3. Determine rdius R 4. Determine the limits of integrtion (Determined y the region) 5. Integrte d. Exmple Find the volume of the solid generted y revolving the region defined y y = x, x = 1 out the x-xis.

4 . Rottion out the y-xis:. Similrly, we cn find the volume of the solid when the region is rotted out the y-xis.. The Volume of the solid generted y revolving out the y-xis the region etween the y-xis nd the grph of the continuous function x=r(y), c<y<d, is V = [ R( y) ] dy. c. Exmple Find the volume of the solid generted y revolving the region defined y y = x 3, y = 8 nd x = 0 out the y-xis. 3. Rottion out line other tht one of the xes:. If you re revolving the plnr oject out one of the lines tht ound the function, then you proceed in pretty much the sme wy, since you still hve solid with no hole. In this type of prolem, the rdius of the disk is still the distnce etween the curve nd the xis of revolution (i.e., sutrct rdius R from the xis of revolution or vise vers, whichever gives positive length).. Exmple Find the volume of the solid generted y revolving the region defined y y = x nd y = 1 out y=1.

5 C. Wsher Method 1. Development of Formul. If the region we revolve to generte the solid does not order on or cross the xis of revolution, the solid will hve hole in it. The cross section perpendiculr to the xis of revolution re wshers insted of disk.. The dimensions of typicl wsher re given: Outer Rdius: R(x) Inner Rdius: r(x) c. Thus the re of the wsher will e given s: A x ( ) = ( outer rdius) ( inner rdius) = R( x) ( ) = R( x) A x ([ ] [ r ( x ) ] ) [ ] [ r( x) ] d. The volume of such solid of revolution would e given s: V = ([ R ( x ) ] [ r ( x ) ] ) dx e. This formul cn e djusted to e integrted with respect to y. f. The disk formul for finding volume is just the wsher formul with r(x)=0.. Exmples. Find the volume of the solid generted y revolving the region defined y y = x nd y = x out the x-xis.

6 . Find the volume of the solid generted y revolving the region defined y y = x, y = 0 nd x = 4 out the y-xis. c. Find the volume of the solid generted y revolving the region defined y y = x +, y = 1 x+1, x = 0 nd x = 1 out y=3.