Department of Mathematics

University of Pittsburgh

MATH 0230 (Calculus II)

Midterm 1, Fall 2017

Instructor: Kiumars Kaveh

Last Name: Student Number:

First Name:

TIME ALLOWED: 50 MINUTES. TOTAL MARKS: 50NO AIDS ALLOWED.WRITE SOLUTIONS ON THE SPACE PROVIDED.PLEASE READ THROUGH THE ENTIRE TEST BEFORE STARTINGAND TAKE NOTEOF HOWMANY POINTS EACHQUESTION ISWORTH.FOR FULLMARKYOUMUST PRESENT YOUR SOLUTION CLEARLY.

Question Mark

1 /10

2 /10

3 /10

4 /10

5 /10

6 /1

TOTAL /50 + 1

1

Kaveh I

Aydin

1.[10 points] Compute the following definite integral. Evaluate and simplifyyour answer: Z 2

0

3x+ 1

(x+ 2)2dx.

2

3×+1

ftp.t#+xBz-z=Acxtz)+B(Xt2)2⇒ 3×+1 = Ax + ( 2A + B) ⇒ A =3 ⇒ 6+B =L ⇒ B= -5 .

) 3,1¥ dx = 3) ×÷zdx- 5) *÷,zd×=

3h ( xtz ) -

5 ( ¥ ) + C

§

"

3¥52dx = ( 34*4+3+-212

=3ln( 4) + § -

3ln( 2) - 5-2 .=3ln(2)-5#

2.[10 points] Compute the following integrals. Evaluate and simplify youranswer.

(a)R10 xe�2xdx.

(b)R 1

01

(1+x2)3/2dx. (Hint: trig substiution.)

3

- ZX

a) Integration by parts : u=×dv= e DX

- ZX

du =d× v = - ez

fudv = uv - fvdu

Sxi "a×=¥i"

.

fE2Ix=±E"i¥+c2

- ZX

= ÷ fxts

foxes"ax- dig

.ES#na+tT=tD- ( by I Hospital ) sees

b) ×=+ana ⇒ d×= sects , do ⇒ 5*2,3,zd×=f -do'sec3D

It x2=sec2( a )

= fsetodotfcassda =sin (a) + C

,

×→⇒⇒o£tzIfkj⇒j←÷×%a×-Csincosb"

.sin . .

sina.rs*

=L 1)o

=D4

3.[10 points] Set up (but do not evaluate) integrals that represents the fol-lowing:

(a) [3 points] The length of the piece of the curve given by the graph ofthe function

y =ex � e�x

2from x = �1 to x = 1.

(b) [7 points] The volume of the solid obtained by revolving the regionenclosed by the curves x = 4 � y2 and the y-axis around the liney = �5.

4

÷¥×j=e±¥'

length = ftp.e#y2dx- ×=4 - y2

b)y=± #

f⇒*÷Ek*t.az. ... # . . - . .

g washersshells

-vol

.

=tf(s+Fx[( S . E)

"

DX cylindrical

washer0method

or

2got.

=D ) ( Y + 5) ( 4- y3dy

Cylindrical -2

shells method

4.[10 points] Find the work done in pumping water out of a circular coneshaped pool which filled up completely. You do not need to simplify orcompute any multiplication of constants like ⇡. The water density is 1000kilogram per meter cubed.

5

tiff10

-

ft }d9a}§¥•

p = weight density of water =1000kg . 91ms

pM C 10 - yjhdy =

weight of the layerat

depth Y

is

10

work =

fpnuo-y5ydya.pnf@oy-zoy2ty3jdyoo-npfoy3.2gI.yIfoD-npfso.o. ngos + zsoo )

5.[10 points] Consider the points P = (1, 1, 1), Q = (1, 0,�1) and R =(2, 2,�1).

(a) Find the area of the triangle with vertices P , Q and R.

(b) Find the equation of the plane that passes through P , Q and R.

6

a)

Area = ± Arpeaanofeeeagnam= 'z I PD ×

PTR I

¥QR

PI = ( 1,

0 ,-1 ) - ( 11 1

, 1) = ( 0,

- 1 ,- 2)

P→R = ( 2/2 ,-1 ) - ( 1 1

1 ,l ) = ( 1 /

1 ,- 2)

PIX PR =det [ oigtgkz ) =

4 F -2T +

k→

1 P§×P→R 1 = if = if → Area a Td .

-

b )F = ( 4 ,

-2 , 1)

4 × - 2y + Z + d =D

4 . 1 -2 . I + 1 + d = o => d =-3

2×-29-2--3=0--1

6.[1 point] Draw a cartoon of yourself writing the test!

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