6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.

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1 6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted y some distnce, h. Whether it is rectngulr prism, tringulr prism, or circulr prism (cylinder), etc., if we cn find the re of the fce, we need only multiply y the distnce etween the two fces, h. In generl V A h For the shpes ove, the re of the fce is the sme t every point sliced prllel to the fce. This is not lwys the cse, nd this is where clculus comes in. (Clculus enters stge left) Imgine slicing lof of red (mthemticlly). It might look like this At ech slice, we hve fce with different re, ut one tht might e function of where the lof ws sliced long the horizontl. For finite slice, the mesurle thickness of the slice, whether it e for sndwich or Texs Tost, is x. This thickness forms the distnce h, etween the two fces. We cn slice up the lof from left to right using uniform x. Pge 1 of 11

2 We cn pproximte the volume of the entire lof y finding the volumes of ech slice V x A x x nd dding them up (This works much etter thn finding volume y the Archimeden method of fluid displcement, which leves the red rther soggy. Incidentlly, Archimedes is clled the Fther of Integrl Clculus since he ws the first person to envision finding volumes y this thin, slicing method). As we slice the regions thinner nd thinner nd thinner, pproching infinitely thin, we lose the ility to sndwich piece of met etween two sliced, ut we lso get incresingly etter pproximtions of the volume. Here s the summry: Definition of Volume Let S e solid tht lies etween x nd x. If the cross-sectionl re of S in the plne through x nd perpendiculr to the x-xis is A x, where A is continuous function, then the volume of S is i x 0 V lim A x x A x dx i i Exmple 1: The volume formuls for the shpes shown t the top of this lesson nd the others from your geometry clss (or relted rte nd optimiztion sections) re derived from clculus. Let s show tht the formul for the 4 3 volume sphere of rdius r is V r. 3 Pge 2 of 11

3 Anytime our cross-sections, perpendiculr to n xis of rottion (or revolution), re circles (or thin cylinders clled discs), we cn us similr pproch. Very often we will hve to crete/envision our solids y rotting or revolving given region round or out n xis. When we crete solids y revolving round n xis tht is perpendiculr to our slices, our cross-sections will lwys e circulr. Disc Method for Volumes of Solids of Rottion When the volume of solid is otined y rotting region perpendisculr to the xis of rottion nd the cross-sections re discs or circles, the volume of the solid is given y Where 2 V R x dx R x is the rdius of rottion s function of x. Exmple 2: Find the volume of the solid formed y rotting the region ounded y the x-xis, y x, nd x 1 round the x-xis. perpendisculr Pge 3 of 11

4 Exmple 3: Find the volume of the solid formed y rotting the region ounded y the y 1, y x, nd x 0 round the line y 1. Exmple 4: Find the volume of the solid otined y rotting the region ounded y the y-xis. 3 y x, y 8, nd x 0 out Exmple 5: Wht if we were to tke the region from the previous exmple nd rotte it round the x-xis insted of the y-xis? Wht would the shpe look like? Wht would perpendiculr slice look like? Find the volume of the solid otined y rotting the region ounded y 3 y x, y 8, nd x 0 out the x-xis. Pge 4 of 11

5 Sometimes, our cross sections re circles ut hve void or hole in them. In this cse, our circulr cross-section, perpendiculr to the xis of rottion, will resemle wsher, with n inner, smller rdius r, nd lrger, outer rdius R. In this cse, the re of the fce of the cross section will e A x R x r x R x r x Wsher Method for Volumes of Solids of Rottion/Revolution When the volume of solid is otined y rotting region perpenwashulr to the xis of rottion nd the cross-sections re wshers, the volume of the solid is given y Where 2 2 V R x r x dx R x is the lrger, outer rdius of rottion nd r x is the smller, inner rdius rottion. Exmple 6: The region in the first qudrnt enclosed y the y-xis nd the grphs of revolved out the x-xis to form solid. Find its volume. y cos x nd y sin x is Importnt things to consider when using the Wsher method: Drw picture, drw picture, drw picture,... You must identify the region 1 st! Like the Disc method, the cross-sections (slices/representtive rectngles) must e PERPENDICULAR to the xis of rottion/revolution Before writing n eqution for R nd r, drw them on your digrm. If you cn drw them, you cn write them. When writing n eqution for R nd r, it will still involve TOP BOTTOM (verticl slice) or RIGHT LEFT (horizontl slice). One of these in ech cse will e the xis of rottion itself. DON T FORGET TO SQUARE EACH RADIUS BEFORE SUBTRACTING THEM. The most common error is to integrte s 2 V R x r x dx. This is WRONG. Keep telling 2 r yourself tht you re sutrcting two seprte volumes: R dx r dx. The nd dx re simply fctored out. Pge 5 of 11

6 perpenwashulr Exmple 7: Find the volume of the solid formed when the R enclosed y the curves y x nd the following xes: 2 y x is rotted out () the x-xis. () the line y 2 (c) the line y 5 (d) the y-xis (e) the line x 1 (f) the line x 17 Pge 6 of 11

7 Exmple 8: The region enclosed y the x-xis nd the prol 2 f x 3x x is revolved out the line x 1 to generte solid of revolution resemling Bundt cke. HOW WOULD YOU PREFER TO SLICE THE REGION? Wht is the consequence of this choice? Is there wy to ccommodte your slicing preference with the xis of rottion/revolution????? We MUST find the volume of this cke? (Wht would hppen if the grph ws rotted out the line x 4 insted?) Pge 7 of 11

8 prshell Shell Method for Volumes of Solids of Rottion/Revolution When the volume of solid is otined y rotting region prshell to the xis of rottion nd the r x nd height h x, the volume of the solid is given y cross-sections re cylindricl shells with rdius V 2 r x h x dx We don t need to worry out holes, since we re only integrting/slicing over the intervl,, prllel to the xis of rottion. For solids generted y revolving region round n xis, we now hve method tht will ccommodte our slicing preference. For such prolems, we choose our slicing method first, not the method. The method is determined y compring your slicing preference to the xis of rottion: PerpenDISCulr/PerpenWASHulr or PrSHELL?? Exmple 9: Find the volume of the solid formed y revolving the region ounded y the grphs of x 0, nd x 1 out the y-xis. 2 y x 1, y 0, Pge 8 of 11

9 So wht if we don t generte our solid y revolving it round n xis? Rememer these guys from the eginning of the lesson? V A h If we cn find the formul for the re of the cross-sectionl fce t ny point long the infinitely thin slice, we cn dd them ll up to find the volume. As reminder: Definition of Volume Let S e solid tht lies etween x nd x. If the cross-sectionl re of S in the plne through x nd perpendiculr to the x-xis is A x, where A is continuous function, then the volume of S is V A x dx Imgine tht concrete sl hs een poured. Upon tht sl, wlls re uilt perpendiculr to the sl. If we cn find the re of the fce of one of these wlls, we kind find the volume of tht pnel, nd thus, the entire house. The sl represents the re of the region enclosed y the curves. If we cn find A x (for slices perpendiculr to the x-xis) or A y (for slices perpendiculr to the y-xis),, with respect to x ( dx eing the infinitely thin width of ech we just need to integrte over the intervl slice). Pge 9 of 11

10 Exmple 10: Find the volume of the solids whose ses re ounded y the grphs of y x 1 nd y x 2 1, with the following cross sections tken perpendiculr to the x-xis. Identify the region tht will e the se, find the points of intersection defining the region, then write n eqution, s x, for the side length of ech cross section. () Squres () Rectngles, height of 2 (c) Rectngles, height is five times the se (d) Qurter Circles (e) Semicircles (f) Isosceles Right Tringles, se is short leg (g) Isosceles Right Tringles, hypotenuse is the se (h) Equilterl Tringles Pge 10 of 11

11 Exmple 11: (Clcultor) An oil spill on the surfce of the wter hs surfce shpe, R, defined y the intersections of the equtions f x sin x nd g x sin x s shown in the figure. The depth of the oil spill t 6 ech vlue of x, mesured perpendiculr to the x-xis, hs depth given y D x 2cos x / 2. Find the volume of the oil spill. Exmple 12: x A region R, defined y the intersections of the grphs of y 5x, y 3, nd y 0, is the se of the 5 solid. For this solid, t ech y, the cross section perpendiculr to the y-xis hs re 2 the volume of the solid. A y y 1. Find Pge 11 of 11