5.1 - Areas and Distances

Transcription

1 Mth 3B Midterm Review Writte by Victori Kl SH 63u Office Hours: R 9:5 - :5m The midterm will cover the sectios for which you hve received homework d feedbck Sectios i your book. No clcultors will be llowed o the midterm. No otes or otecrds will be llowed o the midterm. You will be expected to kow formuls d how to use them, d I hve provided most of them i this review. I hve lso selected prctice problems tht should be similr to problems you hve see i your homework. I highly recommed completig your homework before the midterm. Here re some thigs you should kow how to do: Kow how to fid the geerl tiderivtive of fuctio Kow how to estimte the re uder curve usig left d right edpoits or the upper d lower estimtes Kow how to estimte the re uder curve usig midpoits Kow how to write d evlute Riem sum Kow how to covert Riem sum ito itegrl Kow how to evlute itegrl by iterpretig it s re Kow how to tke the derivtive of itegrl FTOC I Kow how to evlute defiite itegrl FTOC II Kow how to evlute idefiite itegrl Kow how to evlute itegrl usig substitutio Kow how to fid the re betwee two curves Kow how to fid the volume of solid usig the disk or wsher method Kow how to fid the volume of solid usig the cylidricl shells method 5. - Ares d Distces We c estimte res uder curves by subdividig the re ito smll rectgles. The pproximte re of fuctio fx o the itervl [, b] is give by: where x = b A fx i x = fx + fx fx x d x i = + i x.

2 5. - The Defiite Itegrl The Riem sum is give by: A fx i x = b fx dx Sometimes we c evlute the Riem sum without covertig to itegrl. Here re some helpful formule you re expected to kow these for the test!: = i = i = i 3 = + Exmple: Fid the exct re uder fx = x 3 + x + x + o the itervl [, ]. First, we wt to fid x d x i : x = b = = The we plug these ito the bove formul: A x i = + i x = + i = i fx i x 3 3 i + i 3 i xi + x i + x i + x i + i i + i + i 3 + i + i + i + i +

3 = = 5 The bove shows the work d fil swer. We c double check our swer by usig itegrl: fx dx = x x 3 + x + x + dx = + x3 3 + x + x The Fudmetl Theorem of Clculus The Fudmetl Theorem of Clculus hs two prts: I. If F x = gx hx ft dt the F x = fgxg x fhxh x. II. b fx dx = F b F where F is the tiderivtive of f Idefiite Itegrls A idefiite itegrl returs the geerl tiderivtive: fxdx = F x + C where F is the tiderivtive of f The Substitutio Rule If u = gx, the du = g xdx, so we hve: b fgxg x dx = gb g fu du Usully u your substitutio vrible is iside of other fuctio. + = = 5 3

4 6. - Ares Betwee Curves For two fuctios fx d gx o [, b], the re betwee them o tht itervl is give by: A = b fx gx dx Aother wy to remember this tht A = top bottomdx. We c lso itegrte over fuctios of y. The re o the itervl y = c to y = d is give by A = d c right leftdx 6. - Volumes Disk or Wsher Method The volume of solid S with cross-sectiol re A perpediculr to the x-xis is give by: V = Axdx The volume of solid S with cross-sectiol re A perpediculr to the y-xis is give by: V = Aydy Usully A = πrdius, or, if you hve two curves, A = πouter rdius πier rdius. Exmple 6. Exmple 6, pg 36 Fid the volume of the solid by rottig the regio foud by x = y d x = y bout the lie x =. The best thig to do would be to drw grph. The cross-sectiol re is perpediculr to the y-xis, so we will be itegrtig with respect to y. x = is goig to be the ceter of our solid, so the outer rdius is give by + y d the ier rdius is give by + y. The A = π + y π + y, so V = π + y π + y dy =... = π Volumes by Cylidricl Shells The volume of solid rotted bout the y-xis betwee two curves is give by V = πxtop bottomdx About the x-xis betwee two curves is give by V = πyright leftdy

5 Exmple Fid the volume of the solid obtied by rottig bout the y-xis the regio betwee y = x d y = x. A quick grph shows tht y = x is the top fuctio d y = x is the bottom fuctio o the regio from x = to x =. Sice we re rottig bout the y-xis, the we itegrte with respect to x. Thus the volume is give by: V = πxx x dx =... = π Averge Vlue of Fuctio The verge vlue of fuctio fx o the itervl [, b] is give by Study Tips Complete ll your homework f ve = b Study importt formuls usig flshcrds Review otes d homework b fxdx Dt Dump : whe you eter the test, write ll the formuls you kow from memory o the mrgi of your test. This wy you wo t hve to remember those formuls durig the middle of the exm. Get good ight sleep d do t tke the test o empty stomch! Prctice Problems The followig re problems from your textbook. These problems should be similr to oes you hve see o the homework, d should be good review for the midterm...9 #6 Fid f give f t = e t + 3 si t, f =, fπ =...9 #6 Fid the positio of prticle give the followig dt: vt =.5 t, s = #7 Fid the upper d lower sums for fx = + si x, x π with =,, Exmple, pg 375 Use the Riem sum to evlute 3 x3 6x dx #7 Express the followig limit s defiite itegrl o the itervl [, 6]: lim x i l + x i x 5

6 6. 5. #38 Evlute the itegrl by iterpretig it i terms of res: 5 5 x 5 x dx 7. Fid the derivtive of #37 Evlute xe + e x dx # Evlute / / hx = x e x z si z + dz dx. Hit: x si x = x. Evlute t + t t dt t. 5. #8 Evlute si x si x dx. 5. # Evlute Hit: sih x = ex e x, cosh x = ex +e x e x sih x + cosh x dx Exmple 3, pg 9 Evlute x x dx. Evlute t xdx. Hit: t x = si x cos x #39 Evlute si x + cos x dx # Fid the re betwee x = y y d x = y y #9 Fid the re betwee y = cos πx d y = x # Fid the volume by rottig the regio bouded by the give curves bout the specified lie: y = e x, y =, x = bout y = #7 Fid the volume by rottig the regio bouded by the give curves bout the specified lie: x = y, x = y bout x = #6 Use the method of cylidricl shells to fid the volume geerted by rottig the followig regio bout the y-xis: y = x x, y = x # Use the method of cylidricl shells to fid the volume geerted by rottig the followig regio bout the x-xis: x + y = 3, x = y.. Fid the verge vlue of the fuctio o the give itervl: gx = x + x 3, [, ]. 6