Math 182 Test 1

You must show all work!  6 points each
1. Find the area of the region bounded by the graphs:  Please Graph!  y = -x2 + 4x   and    y  = x - 4

Find intersection of function
y
= -x2 + 4x    and   y  = x - 4

-x2 + 4x = x -

0 =   x2 - 4x + x - 4

0 = x2 - 3x - 4

Integrate with respect to -x2 + 4x > x - 4  on [-1, 4] (x - 4)(x + 1) = 0

 

x = 4 or -1


2. Find the area of the region bounded by the graphs:  Please Graph!  x = y2 -3y   and    x = 0


#3-5

 SET UP ONLY

Set up an integral that can be used to find the volume of the solid obtained by revolving the shaded region about the indicated axis

3. y = 0   (x-axis)

Shell Method:

width = dy
average radius =  y
height
= ( 2 - y) - y2

Disk Method: 

router = Öx ,rinner = 0 on [0,1]

router = 2- x ,rinner = 0 on [1,2]


4. y = 1

Shell Method:

width = dy
average radius =
1- y
height
= (2 - y) - y2

Disk Method:

router = 1 ,rinner = 1-Ö on [0,1]

rinner =1 ,rinner = 1-(2-x)  on [1,2]


5. x = 2
Shell Method width = dx
average radius =
1- y
h
1 = Öxh2 = 2 - x
Disk Method: router = 2 - y2

rinner = 2 - (2 - y)


6. Find the volume of the solid formed by revolving the region bounded by   y = 0, y = x3, and x = 2 about the line y = 8.  Use Disk Method

router = 8 - 0 = 8

rinner = 8 - x3

on [0, 2]


7. Find the volume of the solid formed by revolving the region bounded by     about the y-axis.  Shell Method

width = dx

average radius = x

height = 2Öx

 


 
8. Find the volume of the solid formed by revolving the region bounded by 
 y = x
y = 0, and x = 1 about the line y = 1.     Any method.

Shell Method:

Disk Method:

                width = dy
 average radius =
1 - y
               height
= 1 - y

router = 1 - 0 = 1
rinner = 1
- x      


9. Use the Shell Method to find the volume of the solid formed by revolving the region bounded about the
y axis   

                   width = dx
    average radius =
x
             


10. Use the Shell Method to find the volume of the solid formed by revolving the region bounded about the
y  = -2

width = dy
    average radius =
y - (-2) = y +2


11. Find the arc length of the graph of the equation from [8, 16] on the x axis.

12.

Write the definite integral that represents the arc length of one period of the curve 
  Do not solve


13. Verify the following has an inverse and find it.  State the domain and range of each.
Inverse:    on (-¥,-1) È (-1, ¥Undefined at -1
Decreasing  Þ 1 to 1Þ has inverse.


Domain: 
Range:  
f(x)
x
Î R, x ¹ -1  (vertical asymptote)
y Î R, y ¹ 0  (horizontal asymptote)
f -1(x)
x
Î R, x ¹ 0  (vertical asymptote)
y Î R, y ¹ -1  (horizontal asymptote)

14. Find the derivative of

Definition:


15. Find the derivative of  

Definition:

Remember the chain rule

16.   Use implicit differentiation to find y'

 

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