This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 2209809, Summer 2011 Exam 3July 25, 2011 Page 1 of6 Name: 5 6‘ [aim m S Signature: Math 220: Multi—Variable Calculus
Exam 3—— Monday, July 25, 2011 Instructions:
1. Clearly print your name and sign your name in the space above. 2. There are 5 problems, each worth a speciﬁed number of points, for a total of 45 points. 3. Please work each problem in the space provided. Extra space is available on the back of each exam
sheet. Clearly, identify the problem for which the space is required when using the backs of sheets. 4. Show all calculations and display answers clearly. Unjustiﬁed answers will receive no credit.
5. Write neatly and legibly. Cross out any work that you do not wish to be considered for grading. 6. Calculators may be used but all of your answers must be justiﬁed. Also, all derivatives and integrals
are to be found by learned methods of calculus. 7. The following table is strictly for grading purposes. Please do not mark. Math 2209809. Summer 2011 Exam 3July 25, 2011 Page 2 of 6 1. Evaluate 2xy dA, where D is the triangular region with vertices (0, 0), (1, 2), and (0, 3)
D (a) Sketch the region D. (0)3 7’33 ﬁx (b) Write the region D in proper set notation. D:{(w 090.41) 2x4. Yé 3&4?
J;
3AM Above, (0) Evaluate the integral. ___"_—___—ﬂ_,—___—_w__.__—.__—.H.————
Math 2209809, Summer 20] 1 Exam 3July 25, 2011 Page 3 of7 f“ 2. Use polar coordinates to ﬁnd the volume of the solid Wm inside the W Lem/NS Flt/(C,
x2 + y2 + 2:2 = 16 and outside of the cylinder x2 + y2 = 4 .
(a) Sketch the region of intersection in the xyplane. ’Tlnc Sf’laang xl—FYzler—lé I‘mlcrch(s “ILL; X)” (P/anc "La/Lax
25‘0 SQ) X14 725 ’6. I'm XV f/anc w¢ («w/g; UJCZ Wam'l’ Veafam
qéxz—Hﬂé Ié (b) Write the region of integration in proper set notation. (i.e. R = [(r,t9) a S r S b, (x S r S [3]) {Zzé’ﬂ/aﬂ zgrgq) 03 a; 277}
(c) Set up a double integral and evaluate it and give the exact answer. vaﬁ W144 njﬂf teem/6
R 0 , A ‘2 f S‘Hr(/6H(Z)VZCIV 0 7, 3
: elm (/é— r2)%j: Math 2209809, Summer 2011 Exam 3July 25, 201] Page 4 of5 A 3. Find the surface area of the part of the surface 2 = xy that lies in the cylinder x2 + y2 = 1. Lara—é ﬂywhxy. [MtHA XZ—Hzél
fxzy) [Y:X ‘ 277 I I u ' 2. .__ 7“ er a 9
A(g,):ﬁmg,4, £9 55477:]: c
D , 3/ “ii
ISZqTféi' (HAL—L) 2 {to A6
0 ._.,. gjng (zd’Z‘JDJQZZgQJZ—J) +z2 4. Evaluate 6W”: 2a'V , where B is the unit ball B = {(x,y,z)  x2 + y2 + z2 5 1} .
H LeF US~ USQ SFL'CHCC/ti (oo‘rchna‘grj 81;“) 9) g)\ 0454.1) oéaézﬁ‘goéﬂg’rrf Math 2209809, Summer 20] 1 Exam 3July 25, 2011 Page 5 of5 5. Use the transformation x 2 Zn, y = 3v to evaluate the integral x2 dA, where R is the region
R bounded by the ellipse 9x2 + 4y2 = 36. (a) Find the Jacobian of the transformation. 009)”): Z 0) :6
004/ V) O 3
(b) Find the image of the region R under the transformation.
{for Msrﬂavma‘ffon 4X24 LI YZA ﬂ?  " (c) Use the transformation to change the integrand. (d) Evaluate the integral. SS XQcM: (340(2) (6) Jaclv [Z a24uzdl 3&5 224 a? clouJV
Chaij/LO Polar (omeralcs aux/2w 2/“, (jg:\’(059/ VSYZSMOV
S S (M rszeyclfcla 0 _ a Z
3: 2q(5ﬂ(0529 J€)(SIV30/r):2¢l[ll9lzjjsw old;
0 o
: Morel—ﬁe? ...
View
Full Document
 Summer '08
 Helton
 Math, Multivariable Calculus, 2W, 6W, LI YZA, 0529 J, Veafam

Click to edit the document details