BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.2, Problem 45E
To determine

To evaluate : 13ex+2dx

Expert Solution

Answer to Problem 45E

  13ex+2dx=e3(e21)

Explanation of Solution

Given information : f(x)=ex+2

Formula used : abf(x)dx=limxxi=1nf(xi)     when x=ban

Calculation :

   We have, x= 31 n = 2 n  

   Here, f( x )= e x+2    and     x 0 =1 x 1 =1+ 2 n , x 2 =1+ 2 n + 2 n = 4 n

   x 3 =1+ 6 n ,...................& x i =1+ 2i n 1 3 e 2+x dx= lim x 2 n i=1 n f( xi )

   = lim x 2 n i=1 n e ( 1+ 2i n )+2 = lim x 2 n i=1 n e 3+ 2i n

   Now,    i=1 n e 3+ 2i n = e 2 i=1 n e 1+ 2i 2n = e 2 e ( 3n+2 n ) e ( n+2 n ) e 2 n 1

   Now,    lim x 2 n i=1 n e 3+ 2i n

   = e 2 lim x 2 n e ( 3n+2 n ) e ( n+2 n ) e 2 n 1

   = e 2 lim x 2 n e ( 3n+2 n ) e ( n+2 n ) e 2 n 1 2 n

   = e 2 ( e 3 e )= e 3 ( e 2 1 )

Therefore, 13ex+2dx=e3(e21)

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