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1 C Integrtion Volumes PhsicsAndMthsTutor.com. Using the sustitution cos u, or otherwise, find the ect vlue of d 7 The digrm ove shows sketch of prt of the curve with eqution, < <. The shded region S, shown in the digrm ove, is ounded the curve, the -is nd the lines with equtions nd. The shded region S is rotted through rdins out the -is to form solid of revolution. Using our nswer to prt, find the ect volume of the solid of revolution formed. Totl mrks. Using the identit cosθ sin θ, find sin θ dθ. Edecel Internl Review

2 C Integrtion Volumes PhsicsAndMthsTutor.com The digrm ove shows prt of the curve C with prmetric equtions tnθ, sinθ, θ < The finite shded region S shown in the digrm is ounded C, the line nd the - is. This shded region is rotted through rdins out the -is to form solid of revolution. Show tht the volume of the solid of revolution formed is given the integrl k sin 6 θ dθ where k is constnt. 5 c Hence find the ect vlue for this volume, giving our nswer in the form p q, where p nd q re constnts. Totl mrks Edecel Internl Review

3 C Integrtion Volumes PhsicsAndMthsTutor.com. The digrm ove shows prt of the curve. The region R is ounded the curve, the -is, nd the lines nd, s shown shded in the digrm ove. Use integrtion to find the re of R. The region R is rotted 6 out the -is. Use integrtion to find the ect vlue of the volume of the solid formed. 5 Totl 9 mrks. O The curve shown in the digrm ove hs eqution. The finite region ounded the curve, the -is nd the lines nd is shown shded in the digrm. This region is rotted through 6 out the -is to generte solid of revolution. Find the volume of the solid generted. Epress our nswer s single simplified frction, in terms of nd. Totl 5 mrks Edecel Internl Review

4 C Integrtion Volumes PhsicsAndMthsTutor.com 5. Figure The curve with eqution, >, is shown in Figure. The region ounded the lines Figure.,, the -is nd the curve is shown shded in This region is rotted through 6 degrees out the -is. Use clculus to find the ect vlue of the volume of the solid generted. 5 Figure A B Figure shows pperweight with is of smmetr AB where AB cm. A is point on the top surfce of the pperweight, nd B is point on the se of the pperweight. The pperweight is geometricll similr to the solid in prt. Find the volume of this pperweight. Totl 7 mrks Edecel Internl Review

5 C Integrtion Volumes PhsicsAndMthsTutor.com 6. O The curve with eqution, sin,, is shown in the figure ove. The finite region enclosed the curve nd the -is is shded. Find, integrtion, the re of the shded region. This region is rotted through rdins out the -is. Find the volume of the solid generted. 6 Totl 9 mrks 7. e R O The figure ove shows the finite shded region, R, which is ounded the curve e, the line, the line nd the -is. The region R is rotted through 6 degrees out the -is. Use integrtion prts to find n ect vlue for the volume of the solid generted. Totl 8 mrks Edecel Internl Review 5

6 C Integrtion Volumes PhsicsAndMthsTutor.com 8. C, O Figure Figure shows prt of the curve C with eqution, >. The finite region enclosed C, the lines, nd the -is is rotted through 6 out the -is to generte solid S. Using integrtion, find the ect volume of S. 7 C T, R O Figure The tngent T to C t the point, meets the -is t the point,. The shded region R is ounded C, the line nd T, s shown in Figure. Using our nswer to prt, find the ect volume generted R when it is rotted through 6 out the -is. Totl mrks Edecel Internl Review 6

7 C Integrtion Volumes PhsicsAndMthsTutor.com 9. R O The digrm ove shows prt of the curve with eqution 6, >. The shded region R is ounded the curve, the -is nd the lines with equtions nd. This region is rotted through rdins out the -is. Find the ect vlue of the volume of the solid generted. Totl 8 mrks. C R O The digrm ove shows prts of the curve C with eqution Edecel Internl Review 7

8 C Integrtion Volumes PhsicsAndMthsTutor.com. The shded region R is ounded C, the -is nd the lines nd. This region is rotted through 6 out the -is to form solid S. Find, integrtion, the ect volume of S. 7 The solid S is used to model wooden support with circulr se nd circulr top. Show tht the se nd the top hve the sme rdius. Given tht the ctul rdius of the se is 6 cm, c show tht the volume of the wooden support is pproimtel 6 cm. Totl mrks. C R O A Edecel Internl Review 8

9 C Integrtion Volumes PhsicsAndMthsTutor.com The digrm ove shows the curve C with eqution f, where f 8, >. Given tht C crosses the -is t the point A, find the coordintes of A. The finite region R, ounded C, the -is nd the line, is rotted through rdins out the -is. Use integrtion to find, in terms of the volume of the solid generted. 7 Totl mrks. R O The digrm ove shows prt of the curve with eqution c, where c is positive constnt. The point P with -coordinte p lies on the curve. Given tht the grdient of the curve t P is, show tht c p. Given lso tht the -coordinte of P is 5, Edecel Internl Review 9

10 C Integrtion Volumes PhsicsAndMthsTutor.com prove tht c. The region R is ounded the curve, the -is nd the lines nd, s shown in the digrm ove. The region R is rotted through 6 out the -is. c Show tht the volume of the solid generted cn e written in the form k q ln, where k nd q re constnts to e found. 7 Totl mrks. A O The digrm ove shows grph of sin, < <. The mimum point on the curve is A. Show tht the -coordinte of the point A stisfies the eqution tn. The finite region enclosed the curve nd the -is is shded s shown in the digrm ove. A solid od S is generted rotting this region through rdins out the -is. Find the ect vlue of the volume of S. 7 Totl mrks Edecel Internl Review

11 C Integrtion Volumes PhsicsAndMthsTutor.com. R O The digrm ove shows prt of the curve with eqution. The shded region R, ounded the curve, the -is nd the lines nd, is rotted through 6 out the -is. Using integrtion, show tht the volume of the solid generted is 5 ln. Totl 8 mrks 5. M O The digrm ove shows the curve with eqution e. Find the -coordinte of M, the mimum point of the curve. 5 The finite region enclosed the curve, the -is nd the line is rotted through out the -is. Edecel Internl Review

12 C Integrtion Volumes PhsicsAndMthsTutor.com Find, in terms of nd e, the volume of the solid generted. 7 Totl mrks Figure shows pperweight with is of smmetr AB where AB cm. A is point on the top surfce of the pperweight, nd B is point on the se of the pperweight. The pperweight is geometricll similr to the solid in prt. Find the volume of this pperweight. Totl 7 mrks 6. Find the volume generted when the region ounded the curve with eqution, the -is nd the lines nd is rotted through 6 out the -is. Give our nswer in the form ln, where nd re rtionl constnts Totl 7 mrks Edecel Internl Review

13 C Integrtion Volumes PhsicsAndMthsTutor.com. d sin u B du d sin u du M cosu cosu sin u du Use of cos u sin u M cos u sin u u cos u d ± k du cos u M tn u C ±k tn u M cosu u cosu u M tn u tn tn A 7 V d M 6 d 6 integrl in M 6 6 their nswer to prt Aft [] sin θ θ θ θ θ θ C M A. d cos d sin d tn θ sec dθ θ d d dθ sin θ sec θ dθ M A dθ Edecel Internl Review

14 C Integrtion Volumes PhsicsAndMthsTutor.com sinθ cosθ dθ M cos θ 6 sin θ dθ k 6 A tnθ θ, 6 6 V sin θ dθ tnθ θ B 5 6 c 6 sin θ V 6 θ M 6 sin Use of correct limits *M 6 p, q A 8 []. AreR d d Integrting to give ± k. M. Correct integrtion. Ignore limits. A 9 Sustitutes limits of nd into chnged function nd sutrcts the correct w round. M 9 units A Answer of with no working scores MAMA. Volume d Use of V d.. Cn e implied. Ignore limits nd d. B Edecel Internl Review

15 C Integrtion Volumes PhsicsAndMthsTutor.com 9 d ± k n M [ 9 n ] [ 9 9 n9 n ] 9 n A Sustitutes limits of nd Note tht ln cn e implied s equl to. So Volume nd sutrcts the correct w round. dm 9 n9 9 n9 or 9 n or 8 n A oe isw 5 Note the nswer must e one term ect vlue. Note tht 9 n9 c oe. would e wrded the finl A. Note, lso ou cn ignore susequent working here. [9]. Volume d d d Use of V d. Cn e implied. Ignore limits. B Integrting to give ±p A M Sustitutes limits of nd nd sutrcts the correct w round. dm Edecel Internl Review 5

16 C Integrtion Volumes PhsicsAndMthsTutor.com * A ef 5 * Allow other equivlent forms such s. or or or Note tht n is not required for the middle three mrks of this question. Aliter W Volume d d d Use of V d. Cn e implied. Ignore limits. B Appling sustitution u u d d nd chnging limits u so tht nd, gives d u u u u Edecel Internl Review 6

17 C Integrtion Volumes PhsicsAndMthsTutor.com Integrting to give ± pu M u A Sustitutes limits of nd nd sutrcts the correct w round. dm * A ef 5 * Allow other equivlent forms such s. or or or Note tht is not required for the middle three mrks of this question. [5] 5. Volume d 9 d 9 9 d Use of V d Cn e implied. Ignore limits. B Moving their power to the top. Do not llow power of. Cn e implied. Ignore limits nd 9 M Edecel Internl Review 7

18 C Integrtion Volumes PhsicsAndMthsTutor.com Integrting to give ±p A M Use of limits to give ect vlues of or or or ef Aef 5 6 Note: 9 or implied is not needed for the middle three mrks. Aliter W Volume d d 6 d Use of V d Cn e implied. Ignore limits. Moving their power to the top. Do not llow power of. Cn e implied. Ignore limits nd Integrting to give ±p 6 6 A 6 B M M Use of limits to give ect vlues of or or or ef Aef 5 6 Edecel Internl Review 8

19 C Integrtion Volumes PhsicsAndMthsTutor.com Note: or implied is not needed for the middle three mrks. From Fig., AB As units cm units then scle fctor k. Hence Volume of pperweight 6 V cm cm their nswer to prt M 6 6 or wrt 6.8 or or ef A [7] 6. Are Shded sin cos d Integrting sin to give k cos with k. Ignore limits. M [ 6cos ] or 6cos cos A oe [ 6 ] [ 6 ] 6 6 A co Answer of with no working scores MAA. Volume sin d 9 sin d Use of V d. Cn e implied. Ignore limits. M [NB: cos ± ±sin gives sin cos ] Edecel Internl Review 9

20 C Integrtion Volumes PhsicsAndMthsTutor.com [NB: cos ± ±sin gives sin cos ] M Considertion of the Hlf Angle Formul for sin or the Doule Angle Formul for sin cos Volume 9 d Correct epression for Volume Ignore limits nd. A 9 cos 9 d [ sin ] Integrting to give sin ; Correct integrtion k k cos k k sin depm; A 9 [ ] 9 9 or A cso Use of limits to give either 9 or wrt 88.8 Solution must e completel correct. No flukes llowed. [6] 7. Attempts V e d M e e d M needs prts in the correct direction M A e e d M needs second ppliction of prts M Aft MAft refers to cndidtes e d, ut dependent on prev. M e e e A co Sustitutes limits nd nd sutrcts to give... dm [dep. on second nd third Ms] Edecel Internl Review

21 C Integrtion Volumes PhsicsAndMthsTutor.com 6 [ e ] e or n correct ect equivlent. A 8 [Omission of loses first nd lst mrks onl] [8] 8. nwhere V d d, ln M ttempt to Using limits correctl in their integrl: {[ ln ] [ ln ] } B M MA,A M V [ / ln] A 7 Must e ect Volume of cone or vol. generted line B V R V S volume of cone V S / M ln or ln9 A [] 9. Use of V d M 6 6 ; Integrting to otin ; 8 ft constnts onl M Aft; Aft correct use of limits V units A 8 M [8] Edecel Internl Review

22 C Integrtion Volumes PhsicsAndMthsTutor.com. MA d [dependent on ttempt t squring ] B d d; ln M;A ft [A must hve ln term] Correct use of limits: [ ] [ ] [ ] 9 Volume M [M dependent on prev. M] ln or equivlent ect A 7 Showing tht t nd B c Volume nswer to ; 69.5 cm 6 cm * M;A [llow 69 6] []. 8 8 M A M d 8 5 M A M A ft Volume is 5 7 units A 7 [9]. d c d M d Attempt d c When p p c p * A cso Edecel Internl Review

23 C Integrtion Volumes PhsicsAndMthsTutor.com 5 p c nd solve with c p M 5 p p c * A cso c 8 6 ; terms correct M 6 d 8ln some correct M ll correct A 6 d 8ln 8ln6 M Use of correct limits 9 8ln V d ; V 9 8ln V d B k 9; A q 8 A 7 []. d sin sin cos M, A d At A sin sin cos dm sin cos essentil to see intermedite line efore given nswer tn * A V d sin d M cos cos d M A [ ] cos sin sin d M [ ] [ cos sin cos ] A [ ] M [ ] A 7 [] Edecel Internl Review

24 C Integrtion Volumes PhsicsAndMthsTutor.com. Volume d M d d B ln M A A ft Using limits correctl M Volume 8 ln A 5 ln A 8 [8] 5. d e e M A A d Putting d nd ttempting to solve dm d A 5 e d M A Volume e d e d e e d M A e e A ft 6 Volume [ 5 ] 6 e e 6 e M A 7 6 [] 6. Volume d, d M, M [ ln ] A A Edecel Internl Review

25 C Integrtion Volumes PhsicsAndMthsTutor.com [ ln ln ] 6 M A [5.75 ln ] A [7] Edecel Internl Review 5

26 C Integrtion Volumes PhsicsAndMthsTutor.com. Answers to prt were mied, lthough most cndidtes gined some method mrks. A surprisingl lrge numer of cndidtes filed to del with cos u correctl nd mn did not recognise tht d sec d tn C in this contet. Nerl ll cos converted the limits correctl. Answers to prt were lso mied. Some could not get eond stting the formul for the volume of revolution while others gined the first mrk, sustituting the eqution given in prt into this formul, ut could not see the connection with prt. Cndidtes could recover here nd gin full follow through mrks in prt fter n incorrect ttempt t prt.. The responses to this question were ver vrile nd mn lost mrks through errors in mnipultion or nottion, possil through mentl tiredness. For emples, mn mde errors in mnipultion nd could not proceed correctl from the printed cos θ sin θ to sin θ θ cos θ nd the nswer sin θ ws often seen, insted of sin θ. In prt d, mn never found or relised tht the pproprite form for the volume ws dθ d dθ. dθ However the mjorit did find correct integrl in terms of θ lthough some were unle to use the identit sin θ sinθ cosθ to simplif their integrl. The incorrect vlue k 8 ws ver common, resulting from filure to squre the fctor in sin θ sinθ cosθ. Cndidtes were epected to demonstrte the correct chnge of limits. Minimll reference to the result tn, 6 or n equivlent, ws required. Those who hd complete solutions usull gined the two method mrks in prt c ut erlier errors often led to incorrect nswers.. Q ws generll well nswered with mn successful ttempts seen in oth prts. There were few ver poor or non-ttempts t this question. In prt, significnt minorit of cndidtes tried to integrte. Mn cndidtes, however, correctl relised tht the needed to integrte. The mjorit of these cndidtes were le to complete the integrtion correctl or t lest chieve n integrted epression of the form k. Few cndidtes pplied incorrect limits to their integrted epression. A noticele numer of cndidtes, however, incorrectl ssumed sutrction of zero when sustituting for nd so lost the finl two mrks for this prt. A minorit of cndidtes ttempted to integrte the epression in prt using sustitution. Of these cndidtes, most were successful. In prt, the vst mjorit of cndidtes ttempted to ppl the formul d, ut few of them were not successful in simplifing. The mjorit of cndidtes were le to integrte 9 9 to give ln. The most common error t this stge ws for cndidtes to omit dividing. Agin, more cndidtes were successful in this prt in sustituting the limits correctl to rrive t the ect nswer of 9 ln 9. Few cndidtes gve deciml nswer with no ect term seen nd lost the finl mrk. Edecel Internl Review 6

27 C Integrtion Volumes PhsicsAndMthsTutor.com. Most cndidtes used the correct volume formul to otin n epression in terms of for integrtion. At this stge errors included cndidtes using either incorrect formule of d, d or d. Mn cndidtes relised tht the needed to integrte n epression of the form or equivlent. The mjorit of these cndidtes were le to complete the integrtion correctl or t lest chieve n integrted epression of the form p. A few cndidtes, however, integrted to give n epression in terms of nturl logrithms. A significnt minorit of cndidtes sustituted the limits of nd into their integrnd the wrong w round. Onl minorit of cndidtes were le to comine together their rtionl frctions to give n nswer s single simplified frction s required the question. 5. In prt, most cndidtes used the correct volume formul to otin n epression in terms of for integrtion. At this stge errors included cndidtes using either incorrect formule of d or d. Mn cndidtes relised tht the needed to integrte n epression of the form k or equivlent. The mjorit of these cndidtes were le to complete the integrtion correctl or t lest chieve n integrted epression of the form p. At this stge, however, common error ws for cndidtes to integrte to give n epression in terms of nturl logrithms. A significnt numer of cndidtes were unle to cope with sustituting the rtionl limits to chieve the correct nswer of. The vst mjorit of cndidtes were unle to gin n mrks in prt. Some cndidtes understood how the two digrms were relted to ech other nd were le to find the liner scle fctor of. Few cndidtes then recognised tht this scle fctor needed to e cued in order for them to go onto find the volume of the pperweight. Insted, significnt numer of cndidtes pplied the volume formul the used in prt with new limits of nd. 6. In prt, most cndidtes relised tht to find the shded re the needed to integrte sin with respect to, nd the mjorit of them produced n epression involving cos ; so gining the first method mrk. Surprisingl significnt numer of cndidtes were unle to otin the correct coefficient of -6, so there dening themselves of the finl two ccurc mrks. Most cndidtes were le to use limits correctl, though some ssumed tht cos is zero. In prt, whilst most cndidtes knew the correct formul for the volume required, there were numerous errors in susequent work, reveling insufficient cre in the use or understnding of trigonometr. The most common wrong strting point ws for cndidtes to write s sin, 9sin or sin. Although some cndidtes thought tht the could integrte sin directl to give them n incorrect epression involving sin, mn relised tht the needed to consider the identit cosa sin A nd so gined method mrk. At this stge, significnt numer of cndidtes found difficultl with rerrnging this identit nd using the cos sustitution A to give the identit sin. Almost ll of those cndidtes who were le to sustitute this identit into their volume epression proceeded to correct integrtion nd full nd correct solution. There were, however, significnt minorit of cndidtes who used the method of integrtion prts in prt, ut these cndidtes were usull not ver successful in their ttempts. Edecel Internl Review 7

28 C Integrtion Volumes PhsicsAndMthsTutor.com 7. There were mn ecellent solutions to this question ut lso too mn who did not know the formul for finding the volume of the solid. Cndidtes who successfull evluted k e d were le to gin 6 of the 8 mrks, even if the formul used ws d with k, ut there were mn cndidtes who mde errors in the integrtion, rnging from the slips like sign errors nd numericl errors to integrting prts in the wrong direction. An error with serious consequences for most who mde it ws to write e s e ; for some it ws merel nottionl prolem nd something could e slvged ut for most it presented trick prolem! 8. For mn cndidtes this proved to e the most testing question on the pper. This ws ttempted with some degree of success most cndidtes nd mn scored full mrks. Most errors occurred in squring nd simplifing. Some of the integrtion ws not good cndidtes ttempting to integrte frction integrting numertor nd denomintor seprtel, product integrting seprtel nd then finding the product, or squred function squring the integrl of the originl function. In some cses cndidtes filed to recognise tht / integrted to log function. Ver few cndidtes knew the formul for the volume of cone, so further volume of revolution ws often found. Cndidtes working with the incorrect eqution for the line epended considerle time clculting n incorrect vlue for the volume. Mn cndidtes produced nswers tht were clerl not dimensionll correct in, nd hence lost ll mrks. It ws common to see epressions of the form finl volume volume re or finl volume volume re Almost ll of those cndidtes who were le to sustitute this identit into their volume epression proceeded to correct integrtion nd full nd correct solution. There were, however, significnt minorit of cndidtes who used the method of integrtion prts in prt, ut these cndidtes were usull not ver successful in their ttempts. 9. Although there were minorit who thought tht the volume required ws d, this is question for which most cndidtes knew the correct procedure nd it is disppointing to record tht less tht hlf were le to give completel correct solutions. Mn were unle to squre 6 correctl nd the integrl of 6 6 gve difficult, oth 6ln nd eing seen. Clcultor errors were lso frequentl noted when simplifing the finl frctions. These rose minl from the incorrect use of rckets. A few cndidtes gve the finl nswer s n pproimte deciml, filing to note tht the question sked for n ect vlue of the volume.. In generl, the ttempts t prt were good nd there ws lrge numer of cndidtes who scored 6 or 7 mrks. Even with poor squring of it ws possile for cndidtes to gin 5 mrks, which helped mn, ut some ttempts t integrtion were not so kindl looked upon; mimum of three mrks ws ville for cndidtes who integrted the numertor nd Edecel Internl Review 8

29 C Integrtion Volumes PhsicsAndMthsTutor.com denomintor seprtel or who produced ln. The mjorit of cndidtes gined the mrk in prt, ut, surprisingl, correct resoning in prt c ws uncommon. The most common pproch ws to pproimte the support to clinder of rdius 6cm nd height 6cm, so ws seen frequentl 8. In prt errors in indices were seen nd in prt mn found epnding 6 stumling lock. In the integrtion, s epected, integrting ws the mjor source of error. However, for the mjorit, the methods nd formule needed for this question were well known nd there were mn completel correct solutions.. The ke to success in prt ws to relize the need to differentite the curve. Mn weker cndidtes did not pprecite this ut there were mn good solutions to this prt. In prt mn cndidtes did not relize tht the needed to comine the result in prt with c nd circulr rguments tht strted ssuming c nd onl used one of these p sttements were seen. Prt c ws nswered ver well nd mn full correct solutions were seen. The volume formul ws well known nd working ectl cused few prolems. There were few errors in squring, where the 8 term ws missing, nd some thought tht the 6 integrl of ws 6 ln.. Most cndidtes mde some ttempt to differentite sin, with vring degrees of success. sin cos ws the most common wrong nswer. Hving struggled with the differentition, severl went no further with this prt. It ws surprising to see mn cndidtes with correct eqution who were not le to tid up the terms to rech the required result. Most cndidtes went on to mke n ttempt t d. The integrtion prts ws generll well done, ut there were mn of the predictle sign errors, nd severl cndidtes were clerl not epecting to hve to ppl the method twice in order to rech the nswer. A lot of quite good cndidtes did not get to the correct finl nswer, s there were numer of errors when sustituting the limits.. It ws plesing to see most cndidtes ppling the volume of revolution formul correctl. However, lthough the question ws well ttempted, with the mjorit of cndidtes scoring t lest 5 mrks despite errors listed elow, full correct solutions were usull onl seen from the etter cndidtes. Algeric errors rose in epnding ; the most common wrong ttempts eing,, nd. Edecel Internl Review 9

30 C Integrtion Volumes PhsicsAndMthsTutor.com Common errors in integrtion were d ln nd d ln. 5. This question involved differentition using the product rule in prt nd integrtion using prts in prt. It ws nswered well, with most of the difficulties eing cused the use of indices nd the ssocited lger. Some cndidtes wsted time in prt finding the - coordinte which ws not requested. A sizele proportion of the cndidtes misquoted the formul for volume of revolution. 6. No Report ville for this question. Edecel Internl Review