In each problem start with graphing a given function to get an idea what is going on. Youcan use the...

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In each problem start with graphing a given function to get an idea what is going on. You
can use the grapher link http://calculus.sfsu.edu/CalculusI /grapher/.
Explain in words why each of the following integrals is improper.

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In each problem start with graphing a given function to get an idea what is going on. You can use the grapher link http://calculus.sfsu.edu/Calculus /grapher/. I Explain in words why each of the following integrals is improper. R 2 8 7 -t (a) t e dt. 2 R p/2 (b) sectdt. p/4 R 3 dt (c) . 1 |t-2| R 2 (d) ln(t-1)dt. 1 1.1 0.9 (a) Graph y = 1/t and y = 1/t . Select two viewing windows (a) [0,10]×[0,1], (b) [0,100]×[0,1]. R R 10 10 dt dt (b) Evaluate and . 1.1 0.9 1 t 1 t R R 100 100 dt dt (c) Evaluate and . 1.1 0.9 1 t 1 t R R 8 8 dt dt (d) Find if it exists. What does this integral represent? Find if it exists. 1.1 0.9 1 t 1 t Determine whether each integral is convergent or divergent. If it is convergent evaluate it. R 8 -2x (a) e dx. 1 R 2 dz (b) . 2 1 (z-1) R 8 dx v (c) . 1 x R 1 Explain why xlnxdx is improper. Evaluate it using the limit of a proper integral. 0 Calculate work needed to take an iphone 5 from the surface of the Earth to in?nity. Google needed constants and parameters. 1

() In each problem start with graphing a given function to get an idea what is going on. You can use the grapher link http://calculus.sfsu.edu/CalculusI/grapher/. Explain in words why each of the following integrals is improper. (a) ∫∞ 2 t7e−t 2 dt. (b) ∫ π/2 π/4 sec t dt. (c) ∫ 3 1 dt |t−2| . (d) ∫ 2 1 ln(t− 1)dt. (a) Graph y = 1/t1.1 and y = 1/t0.9. Select two viewing windows (a) [0, 10]× [0, 1], (b) [0, 100]× [0, 1]. (b) Evaluate ∫ 10 1 dt t1.1 and ∫ 10 1 dt t0.9 . (c) Evaluate ∫ 100 1 dt t1.1 and ∫ 100 1 dt t0.9 . (d) Find ∫∞ 1 dt t1.1 if it exists. What does this integral represent? Find ∫∞ 1 dt t0.9 if it exists. Determine whether each integral is convergent or divergent. If it is convergent evaluate it. (a) ∫∞ 1 e−2xdx. (b) ∫ 2 1 dz (z−1)2 . (c) ∫∞ 1 dx√ x . Explain why ∫ 1 0 x ln x dx is improper. Evaluate it using the limit of a proper integral. Calculate work needed to take an iphone 5 from the surface of the Earth to infinity. Google needed constants and parameters. 1

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David · Accepted Answer

1) An improper integral is the limit of a definite integral as an endpoint of the 
interval(s) of integration approaches either a ‘specified real number’ or ‘∞’ or 
‘−∞’ or, in some cases, as both endpoints approach limits or in simple words 
An integral having at least one non-finite limit or an integrand that becomes 
infinite between the limits of integration
   a)∫        
 

 
  
  
 From the definition of improper integral, the above integral is improper 
since it has upper limit (=infinity), a non- finite one.
   b) ∫          
   
   

  
 
From the definition of improper integral, the above integral shown is improper 
because at      , function vanishes or the integrand becomes infinite. 
  t=2 
?  ∞ 
? 4 
?   
 c)∫         
 
 

From the definition of improper integral & graph shown, the above integral is 
improper because the integrand becomes infinite at    . 
d)∫         

Clearly from the definition and the graph, the above integral is improper 
because the integrand or function becomes infinite ( ∞  at     
2) Given functions are   
 
    

    
 and graphs are plotted below.
At t=1, function dominance changes. For    , 
 
    

    
 and for    ,
    

    
.  
a) Viewing window:   [    ]      [   ]
b) Viewing window: t   [0,100] and y   [0,

In each problem start with graphing a given function to get an idea what is going on. You can use the grapher link http://calculus.sfsu.edu/CalculusI /grapher/. Explain in words why each of the...

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