Hi,

I need step by step solutions to for all the questions in the attached file. Please follow all the instructions as given in the file.

Thanks :)

Sierra College – Math 31 – Exam #1
NAME: ______________________________________ score: /75
You must have read, signed and submitted the exam contract before taking this test. As a reminder, be certain to
show all of your work for each question in order, in a neat and organized manner with explanations, using proper
mathematical notation and presenting a logical argument that justifies your conclusions. Do not share these
questions in any way with anyone. The only resources allowed are the textbook, your handwritten notes, and a
basic calculator. You many not use any type of cell phone or computer (online or offline). You do not need to
write any answers on this page, just include your work on your own paper.
(5points)

  1. Consider the curve
    3
    y x x = − 2 3 . The horizontal line
    y p =
    intersects the curve twice in the first quadrant
    creating two regions bounded by both equations. Find p so that the areas of the two regions are equal. You
    must provide a detailed sketch clearly showing your “representative rectangles” and include all Calc I graphing
    techniques.
    (5points)
  2. Find a horizontal line y = m, so that when the region bounded by the curves
    y x = +1 , y x = −1
    , and y = 0 is
    revolved about the line y = m, the resulting solid has volume
    12 . You must provide a detailed sketch clearly
    showing your “representative rectangles”.
    (5points)
  3. Find the volume of the solid whose base region is an isosceles right triangle with hypotenuse length 2b and
    cross-sections are squares whose sides are parallel to the hypotenuse of the base region.
    (10 points)
  4. Consider the curve
    1
    y x tanh−
    = .
    a. Find the area below this curve in the first quadrant. You must provide a detailed sketch clearly showing
    your “representative rectangles”.
    b. Find the volume of the solid obtained by revolving this region about the line
    x = −1 .
    (5points)
  5. Find the average value of
    2
    1
    f x( )

x

on an interval of positive numbers [a,b]. Show that the value of c in the
conclusion of the Mean Value Theorem for Integrals is
c ab =
. This is called geometric mean of two
positive numbers a and b.
(10 points)

  1. The tank shown below has the shape of an inverted square pyramid and it is full of a liquid with density
    3
    1
    5
    kg
    m
     =
    . Assuming we are on a planet that has acceleration due to gravity of
    10 2 g m

s

, find the work
required to pump all the liquid out of the top of the tank.
(5points each)

  1. Evaluate each of the following integrals. As with all questions, you must fully explain every step in your
    process showing all work and using proper notation. Any missing work or justification will not get credit.
    a.
    ( ) ( )
    2
    3 2 2
    0
    32 sin cos x x x dx

        
    b.
    ln 4
    2
    0 9
    t
    t
    e
    dt
    e +

    c.
    7
    1
    dx
    x x −

    d.
    5
    1
    x 1
    dx
    x

    −

    e.
    1
    2 4
    0 0
    3( 1) 1 ( 1)
    x
    x t dt dx
     
    − − −      
    f.
    2 2
    1
    sinh cosh
    p
    dx
    x x

    +

    where
    1 3 ln
    2
    p
      +
    =      
    g.
    5
    2
    2
    f x dx ( )



    , given that
    cos ( ) x
    f x
    x

where
2
2
f
  
  =
 
and
5
4
2
f
  
  =
 
h. ( )
1
0
x f x dx ( ) 

given
( )( )
2
10 ( ) ( )
( ) 1 4 [ ( )]
g x f x
g x g x

− +
and 2
(0) 2, (1) 0, and (1) ln
9
g g f  
= − = =  