78 Part I: The Questions
Using Limits and Riemann
Sums to Evaluate Definite
Integrals
541–545 Use the limit form of the definition of the
integral to evaluate the integral.
541.
1 2
0
2
+
(
)
∫
x dx
542.
1 3
3
0
4
+
(
)
∫
x dx
543.
4
1
4
−
(
)
∫
x dx
544.
2
4
2
0
3
x
x
dx
− −
(
)
∫
545.
x
x
dx
2
1
3
5
+ −
(
)
∫
Finding a Definite Integral
from the Limit and Riemann
Sum Form
536–540 Express the limit as a definite integral.
Note that the solution is not necessarily unique.
536. lim n i
n
n
i
n
→∞ =
+
( )
∑
3 4 3
6
1
537. lim
sec
n
i
n
n
i
n
→
=
( )
∑
∞
π
π
3
3
1
538. lim n i
n
n
i
n
→∞ =
+
( )
∑
1 6
1
539. lim
cos
s in
n
i
n
n
i
n
i
n
→
=
( ) + ( )

 

 
∑
∞
π
π
π
2
2
2
1
540. lim n i
n
n
i
n
i
n
→∞ =
+
∑
5 5 125
3
3
1