Calculus 2 test

Math3B Exam #02 Solution Spring 2021

1. Integrate.

a) 2

ln e

e x dx³ b)

ln

1 2

e x dx³

u do a lnrx I

4 c

e Z eZ

Xenale JeE dx xenrx E Iee Gente EI CeeEE E E E TEI 2 Ee Z Z

2. Integrate.

a) � �cos ln x dx³ b) ln 8

ln 3 1 xe dx�³

LEE telex x et dt IxDx X dt dy etdt

dxfcoslle.de t etde cost d

Nowby parts u du

Ct Cost et

s sin E

Ct I cost et

S etcosEdt etcost tetsint fetostdt

Ietcostde et cost ts int te Eccost tSint

cfetcostde z

xkos6nxztsinClnxD

3. Integrate. � � 1

2 1

0

2 sin x dx�³

u dir Ct sink

I

C 26in x I_EE

I

f4s xtdx xlsirixflo 2fotsiu ix.EEdxtsiu x 7X sint dt dx

E I t in't II 4 0 t.es w 0

0

It By Parts xGwx5I 21Gt sine de n du

Sint trust7 O Sint

I 6 2

Ils I o 2EtcosttsintIF L 2 If co EtsE CoD36 2E r I t or

Tc2 RjE

4. Integrate. For part b) use trigonometric substitution.

a) 21

lnf ³

x dx

x b)

2

1 3 6 13

dx x x� �³

5. Integrate. 2 0

2sin cos 2 cos x x dx x

S

�³

2

2

2

2

If tan , 2

2 sin

1 1

cos 1

2 1

x u

u x

u u

x u

dx du u

� �

�

�

6. The flat surface shown is submerged in a fluid. Find the fluid force against the surface. The fluid

density is 310 lbs ft

. Note: � � � � b

a F h y w y dyU ³

2 ft

4 ft 5 ft

3 ft were 4 Jd

y x

dy Caldor X 34y

o k

v d 6 y

E fabphlywcydy

10410 G y Fy dy Heightdy

Depth d TLensth x

Eff G y dy 64 y 5 dy

Ez 3y2 33104

Ez448 Et o

Ez 64

5 40 Zoofe