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 = x^{2} + 4x and y = x  4  
Find intersection of function y = x^{2} + 4x and y = x  4 x^{2} + 4x = x  4 0 = x^{2}  4x + x  4 0 = x^{2}  3x 
4 

Integrate with respect to x x^{2} + 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 = y^{2} 3y and x = 0  


#35 
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 (xaxis)  
Shell Method: 
width = dy average radius = y height = ( 2  y)  y^{2} 

Disk Method: 
r_{outer}
=
Öx ,r_{inner
}
= 0 on [0,1]
r_{outer} = 2 x ,r_{inner } = 0 on [1,2] 



4.  y = 1  
Shell Method: 
width = dy average radius = 1 y height = (2  y)  y^{2} 

Disk Method: 
r_{outer}
= 1 ,r_{inner
}
= 1Öx
on [0,1] r_{inner }=1 ,r_{inner } = 1(2x) on [1,2] 



5.  x = 2  
Shell Method  width = dx average radius = 1 y h_{1 } = Öx , h_{2}_{ } = 2  x 

Disk Method:  r_{outer}
=
2 
y^{2} r_{inner }= 2  (2  y) 


6.  Find the volume of the solid formed by revolving the region bounded by y = 0, y = x^{3}, and x = 2 about the line y = 8. Use Disk Method  
r_{outer}
= 8  0 = 8
r_{inner }= 8  x^{3} ^{on [0, 2]} 



7.  Find
the volume of the solid formed by revolving the region bounded by
about the yaxis. 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 
r_{outer}
= 1 
0 = 1 





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 




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 


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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