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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 =
[email protected]. 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!
7
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