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HW7 - Math 17C Kouba Worksheet 2 1 Let R be the region...

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Unformatted text preview: Math 17C Kouba Worksheet 2 1.) Let R be the region bounded by the graphs of y = x3. and y = 4:1: (in the first quadrant). a.) Describe R using vertical cross-sections. b.) Describe R using horizontal cross-sections. 2.) Let R be the region inside the circle of radius 2 centered at (0, O) and above the y-axis. a.) Describe R using vertical cross—sections. b.) Describe R using horizontal cross-sections. 3.) Let R be the triangular region with vertices (0,0), (—2, 3), and (2, 3). a.) Describe R using vertical cross—sections. b.) Describe R using horizontal cross-sections. 4.) Let R be the region bounded by the graphs of m z y2 and a: = 2 — a.) Describe R using vertical cross—sections. b.) Describe R using horizontal cross—sections. 5.) Sketch each of the following regions described in two-dimensional space. a..)0£x§1,3m§y£m+2 b.)0§m§ln4,ezgys4 0-)1SyS4,—flSxS\/fl d-)1SySln5.lnySmSey 6.) Evaluate the following double integrals. 3 1 a)/ /(1+4:Ey)drcdy b.)/01/12y ——dyda: 1 0 2 1 8 1 1 my 2$+y dmdy d. / / ———-————~dyd$ )/0/0( ) )0 0 \/$2+y2+1 1 (1:2 1 e“ ) / / (:c+2y)dydrc f.) / / fidmdy 0 0 0 y 7r/2 cos y . 7r/4 7r/6 ) f / esm y dw dy h.) / / cos 336 sin 23/ dy dac 0 0 0 0 (Bewaie of the next two.) 1 1 i)/01:/je ””2 dwdy j.)// \/:1:3+1d.73dy 3y 0 fl 1 7.) Find the area of the region a.) in problem 1. b.) in problem 4. 8.) Consider a mountain range above the grid 0 S :1: S 5, O S y S 8, where distance is measured in miles. Elevation (miles) above sea level at the point (my) is given by H(:r, y) = (1/40)(10 —— 3:2 + yz). a.) Find the elevation at the points (0,0), (4, 2), and (0,8). b.) Compute the average elevation of this mountain range. 9.) A flat plate lies in region R bounded by the graphs of y = (/5 and y z (1 /2)r Temperature at point (a), y) is given by T(m, y) = 50 + 2x + y (OF). a.) Find the area of the plate. b) Find the average Width of the plate. c. ) Find the average height of the plate. d. ) Find the temperature at the points (0,0), (2, 1.1), and (4, 2). e.) Find the average temperature of the plate. )Sketch the solid 1n 3D- Space whose volume 1s given by the following double integral. y(1/3 10/01/0y(4——:r2——y2)davdy 1.) A flat plate lies 1n region R bounded by the graphs of m— —— 0, y = m3 , and y: $2 + 4. Density at point (11:, y) is given by 6(30, y)— ~ 1 + ."c + 2y grams per square centimeter. a.) Find the area of the plate. b.) Find the average width of the plate. c.) Find the average height of the plate. d.) Find the density at the points (0,0), (1, 3), and (2, 8). e. ) Find the average density of the plate. f.) Find the total mass of the plate. 12.) Compute the volume of the solid lying above the region bounded by the graphs of y = 220, y = 2, and CE = O and below the paraboloid z 2: .732 + 312. 13.) Compute the volume of the solid which is above the region bounded by the graphs of 342172, y=0, andm=2 and between the planesav+y+z=6 and$~y+zz 12. w- , ’ , , ~ \‘i’iijgii;"ii;:::i‘“' *(o ,, l __ f;If:..¥.j;T.[.";_ILZZL,0.i???ZEEEEU 0‘51 0% 2f;,QQQ;‘::21:i:if :: W AAAAAAAA 4"“ ______ W 1 y: z W , 3XEXC><2+W+IJQD o I , {"0 YT; {- .{DW 3“Y 4 X- < 3 :5 TE: < 33 NZ 3» :3” 5'2 ’ V _ ngf 3 >< ? WC 5_ (A) l T g H 0 K O C K 6% 5:1: A) T E @ K A T I 0/U ) WW 3: ng 0‘3 Jar 0‘32 E C e; y y: 3 /’ T0401): 50+8+ék T6340] .. 3. 50,. 0. PM _...__....__.._..,._ww,.. W 6.--...“ 'T’CAJLI) :: {0444/14 —— ~— ~— .- MAM: —_——- I: S W «Wmhw {507+02><7+ifi- 7°94: i 3 GOE iqSfiwébWikEYMgzw- H qu m...__,,.--_...._m..umxhfim0wmmmwmmmm JO 3]~ MM, 8 W Am}: 0%; 8)— maue m W/WRD :2. X144! ) AV€~———— 3 3x3 0+><+azvjog< mg jig-*3 CY+XY+Y°ZJvax3 : 257—3 [Onto]; XCX +4)+ CK +4] —— Cx +x"+><6)] M 3 [x 4.4/4- "AL qx—L + 3x 2.1.](0 VJ)?“ X‘ '3 [fiX‘z-LaL/X4—aog €141 Elli; ”(y ,_ 4:016X—7 __ X7) [A 3 ,. ”Tjjiy330*““§” :m:r.._,. wwwwwwww wfiégmxl{g_xxy}a;f wk? n O fjjjjj} a [2; $1173 ,3 '7,ii’ifwf,,if"ff,"_*§,"l177,,",f,if: -E‘So CQLY‘RD B'F ,9, 3* ., , ,. N m 9» w m : ”itiw?f3%)“ w w- ~ ~ m - m ;:,_,_.5::,>,:w i 1 I! aim W W W ”W” W W W W “FM“ 5 "1— 5’ 5' 0 WWW ...
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