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MATH 1230 Worksheet 10 – area, average, IBP, improper

1.           Consider the area of the region bounded by the curves  = √ ,  = 0 and  = 4. Set up an integral that represents the area of the region. Click for a sketch.



2.             Consider the area of the region bounded by the curves  = √ and  =  . Set up an integral that represents
the area of the region. Click for a sketch.



3.           Find the area of the region bounded the curves  =  2  and  = 0 between  =  and  = 1. Click for a sketch.




4.           Find the area of the region bounded the curves  = 2 −  2  and  =  .Click for a sketch.




5.           Find the area of the region bounded the curves  = 0 and  =  3  −  . Click for a sketch.




6.           Find the area of the region bounded the curves  = ln(),  = 1 and  = 1. Click for a sketch. Hint: requires IBP




7.           Find the area of the region bounded the curves  = 62  − 31 and  = 5 − 32 . Click for a sketch.




8.         Find the area of the region bounded by  =  and  = 6 −  . See sketch.




9.           Find the area of the region bounded by  =   and  = − 2 .Click for a sketch.




10.          Evaluate the following integrals:
(a)    ∫  2                                         (e)    ∫  ln                           (h)    ∫   ln()  



(i)    ∫  5   3      [hint: substitute first]



(c)       5  


(j)    ∫      [hint: substitute first]



(d)   ∫ (ln())2                            (k) ∫       (Hint: use IPB with   = ( − 1)  and   =  − 2 )




1
(m) ∫  



11.       Evaluate the following improper integrals:
(a)        



(g)        −



(j)   ∞    3 − 5||  



(b)



(e)       − 5


               (h)    ∞      


(k)  ∞    3 − 5||  


(c)        



(f)    ∞    5  


(i)    ∞    − 2   



12.  Determine the average value of () =   from x = 1 to x = 5.




13.  Determine  ()   if () = {3



14. Determine  ()   if () = {2()  for   ≥ 5      where  () = 10



15. Determine  ()   if () = {    where  () = 10



16. Determine  ()   if () = {    where  () = 100



17. Determine the value of k if   () = 25   for () = {         for   ≥ 5



18. Find the area bounded by the curves y=f(x) and y=0, for all x>0 if () = {   for   < 2    for   ≥ 2



19. Find the area bounded by the curves                                                                               8 − 2   for   ≥ ln(2)
Hint:  See sketch below




20.  Find the average value of () =   from x=1 to x=2




21. Find the average value of () =   from x=1 to x=e



22a.  Find the average value of f(x) from x = 3 to x = 10 if  () = ∫ ()  and F(10) = 21+C and F(3) = 7+C




22b.  Find the average value of f(x) from x = -3 to x = 3 if  () = ∫ ()  and F(10) = 24+C and F(3) = 48+C




23. Find the average value of f(x) from x = -ln(2) to x = ln(2) if () = 16 − ||





24. Find the average value of f(x) from x = -ln(4) to x = ln(2) if () = 16 ||





25. Find the average value of f(x) from x = -2 to x = 4 if () = {



26. Find the average value of f(x) from x = -1 to x = 2 if () =    where  () = 10




27. If the average value of f(x) from x= 5 to x = 10 is 100, determine ∫()




28. If the average value of f(x) from x= -10 to x = 10 is 100, determine 0 ()




29. If the average value of f(x) from  = − 1000 to  = 1000 is 0, determine   ()



30. If the average value of f(x) from  = 0 to  =  is 100, determine   ()




31. If the average value of f(x)=2x from  = 0 to  =  is 100, determine ℎ () 



32. If the average value of f(x)=3x2 from  = 0 to  =  is 100, determine ℎ ()  




The below are extra/optional questions, meant to prompt some thinking and discussion.
Extra 1. If the average value of f(x) from  = − ∞ to  = ∞ is 0, what can you say about  ∞ ()  ??    

Extra 2. If the average value of f(x) from  = − ∞ to  = ∞ is 100, what can you say about  ∞ ()  ?? 

Extra 3. If the average value of f(x) from  = − ∞ to  = ∞ is ∞, what can you say about  ∞ ()  ??