The standard equation for an ellipse

Math3B Exam #01 Solution

1. Evaluate the integral. Exact answers only. No decimal answers!

a) ( ) 2

1

1 3 5

dx x−∫ b)

8

2

2 ln

e

e x dx

x∫

 

 

 

 

 

 

 

U 3 Jx u lnMx du Sdx

da dx da Tedx

X 2 u 3 lo du _E E dy

u 7 zdu dx X I U 3

J

u z x e8_ u lnfE lae4 4

X eI u lnrE ke I

Isf da zS da

S du z enulI o

lnlull.ie 2 lait AI

Lal 21 Int d 2ln4 lae6

Luz la

lace S

 

24

4G

2

41h4 In 256

In E 5

 

 

 

2. Evaluate the integral. No decimal answers!

a) ( )6 0

2sec sec tanx x x dxe π

∫ b) ( ) 2 2

2 3 4x x dx

− − −∫

a 2 4 ErExidx SI dx u ZsEcxtanxdx oddFunction

Geometrically

Izdu sEcxEauxdx Evaluted

E u 2s cE 4 Half a circle

of Radius 2

o U ZSEC 0 2 3 Zit

413

Ef en du Get 415

Ee4

E e ed

 

 

3. Find the average value of the function on the given interval. No decimal answers!

[ ]( ) 0, 4f x x=

 

 

 

 

 

 

 

 

 

 

 

 

4. Find the area of the region bounded by the curves cos and siny x y x= =

on the interval 5,4 4 π π .

 

 

 

 

 

 

 

fave_Lafabfaddy

fix de 6 8

g E 4312

Y y cosx yesinx A ink cosx dy 5T

o l l l l 1 s cosx siwxl.IT IT 4

X 4 4 5

If cosxtsiNXle 4

f Etf Ed Et ET a a

Ezra ZE

 

 

5. Answer the following questions. Only setup the integrals!

a) The elliptical tank below is full of heating oil with density 100 lb/ft3. Setup the integral that

calculates the work required to pump all the oil out the outlet. Show a picture of the graph you

are using to model the cross-section of the volume and define its equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Note: The standard equation for an ellipse

related to this problem is 22

2 2 1 yx

a b + = .

8 ft

6 ft

10 ft

2 ft

ye

5

fr D 5 y 3

6 16 Ay Bookcase I 7 4 X

3

a zt LI Vs 2x lo Ey

20k1g 42 It If I Vsi 2014 1 sya Ag ft

L GI weight _Vol P 62 16 1 54 2014ft

Ha ay 100 lb

x 4fEE 2000 474 ay 8000 JI TAI Ay

Distance D 5 y

8000 ETCS y dy

W f zooo 47 8 5 y dy

 

 

 

b) The tank below is full of heating oil with density 50 lb/ft3. Setup the integral that calculates the

work required to pump the oil out the outlet. Show a picture of the graph you are using to

model the cross-section of the volume and define its equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 ft

4 ft

3 ft

8 ft

3 ft

 

y r 9 4 2

4x ytz 11 i 4ytE

d 11 y 8 E 8soso.netententes Estacado 4 Ez4

VS lice L w h

8 2x ay 0

I 7 16KEY E E 16C y E ay

Z

Vscice y t 8 ay ft

weight Vol P Distance D 11 y

1614g 8 Ay 50 lb

Weight 80014g 8 dy

work 80054814g t 8 Ie y dy

work 3200148 gtz IL y dy

 

 

 

b) The tank below is full of heating oil with density 50 lb/ft3. Setup the integral that calculates the

work required to pump the oil out the outlet. Show a picture of the graph you are using to

model the cross-section of the volume and define its equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 ft

4 ft

3 ft

8 ft

3 ft

y 4x x t y b r n T o

d 7 y

ajtt 4 asoka ayathande

e E

C o

X L w h v

Stice v 8 2 3 ay b

8121 g t 3 ay

Vsa 8 It 3 ay

weight Vol p 8 It 3 Ay 50 lb

400 It 3 ay

Distance D 7 y

work So400C It 3 7 y dy

work 40064 Et 3 7 y dy

 

 

6. The tank below is full of heating oil with a density of 100 lb/ft3. First set up is an integral that represents the

work required to pump all the oil out the outlet of the tank. Then evaluate the integral to find the work

required to pump all the oil out the outlet.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 ft

6 ft

6 ft

2 ft

y Ex is X Zzy

2 i

d z y o 14 I

not afforded y roasted

Slice L w h G

6 x ay j b

Gl Eg ay ice 4gAy ft3 Distance d

Z y

Weight Vol p 4gAy 100 lb

Weight 400gdy lb

work L 400g Z T dy 400C 108 400 2g 5 dy 43,200 ft lb

E i J

400 36 72

 

 

6. The tank below is full of heating oil with density 100 lb/ft3. First set-up an integral that represents the

work required to pump all the oil out the outlet of the tank. Then evaluate the integral to find the work

required to pump all the oil out the outlet.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 ft

6 ft

6 ft

2 ft

 

y n y Ext6

Exf 6 g E 8 a 4 Ey

G d 8 y

X Befog

Ay Beauteous

C o I 7

slice _low h 4 X

6 X ay

6 4 Ey Iy Work 400 46 y 8 y dy 4 4g ay 4oofo6 48 l4yty4dy

Us 416 DET ft 4oo 48y 7y2tT lo Weight _Vol P 216

416 g Ay 100lb 4001288 252 t 3

weight_4006 51 9 400 36 22

Distance 4 8 y 400408

43,200ft lb

 

 

7. A solid has an elliptical base with equation 2 2

2 1 1x y+ = . Parallel cross-sections perpendicular to the y-

axis are squares. First setup the integral that represents the volume of the solid. Then evaluate the

integral to find the volume of the solid.

 

 

 

 

 

 

 

 

 

 

 

 

x

I1 X X Hy

ZX XZ 2

t y L fz

XZ2x z I YZ I

Ki 4 2 ay

Xd Z zyz

t Usc 4 2 25 IS V 412 25 dy

25 4 z 25 dy

21114 z I y4 dy

16 14 ya dy IG y EsIff 16 H E o

16 E 3 2 3

 

 

8. Answer the following questions.

 

a) The region shown in the figure to the right is bounded

by the graphs ( )22 , 4, and 4y x y x x= − = + = .

Set up the integral that represents the area of the

bounded region by integrating with respect to x.

 

 

 

 

 

 

 

 

 

 

 

 

 

b) The region shown in the figure to the right is bounded

by the graphs ( )22 , 2, 4, and y 6y x x x= − = = = .

Set up the integral that represents the area of the

bounded region by integrating with respect to y.

Note: You will have to use the sum of two integrals

to get the correct answer.

 

 

 

 

 

x-axis

y-axis

y-axis

x-axis

4

A fo4 crxt4 x 25 d 4

4

6 A y 6

4 9 6232 My X z x rft2

WE NEED a function of y HERE X My 12 z 14

X z 6 4

A fo4 crytz 2 dxtf4 4 2 dy

 

 

9. The shaded region shown in the figure below is bounded by the

graphs , 3, 1 and 4y x y x x= = = = . Find the volume V of the solid obtained by revolving the region

about the line 5x = using cylindrical shells.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x-axis

y-axis

O h 3 ix r

3 r n r s Us Zerh AX h

y rx Vsi 2e 5 x 3 F IX IX

I 4 5 z J C5 x C3 re dx

I 4 4 2 4115 5 42 3 812 dx

2e I5x EXk Exit x

2E 60 85 24 651 15 E Etz

2E ZI Et E

z 630 20030372

45

2 342 5

347in 15

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