Part II: The Answers
346
Answers
501–600
531.
lim
n
i
n
n
i
n
→
=
( ) +
∑
∞
3 1 3
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆ = − =
x
n
n
4 1 3
You also want to select a point from each interval. The formula x i = a + (Δ x)i gives you
the right endpoint from each interval. Here, you have
x a
x i
i
n
i = +
= +
( )
∆
1 3
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
lim
n
i
i
n
n
i
n
n
i
f x
x
i
n n
n
i
n
→∞ =
→∞ =
→∞ =
∑
∑
=
+ ( )
=
( ) +
∆
1
1
1
1 3 3
3 1 3
n n
∑
532.
lim
sin
n
i
n
n
i
n
→∞ =
( ) ( )
∑
π
π
2
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆x
n
n
= − =
π
π
0
You also want to select a point from each interval. The formula x i = a + (Δ x)i gives you
the right endpoint from each interval. Here, you have
x a
x i
i
n
i
n
i = +
= +
=
( )
∆
0 π
π
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
sin
lim
n
i
i
n
n
i
n
n
f x
x
i
n
n
n
→∞ =
→∞ =
→∞
∑
∑
=
( )
( )
=
∆
1
2
1
π
π
π
( ( ) ( )
=
∑ sin
2
1
πi
n
i
n
533.
lim
n
i
n
i
n
i
n n
→∞ =
+
( ) + +
( )
∑ 1 4
1 4
4
2
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆x
n
n
= − =
5 1 4
346
Answers
501–600
531.
lim
n
i
n
n
i
n
→
=
( ) +
∑
∞
3 1 3
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆ = − =
x
n
n
4 1 3
You also want to select a point from each interval. The formula x i = a + (Δ x)i gives you
the right endpoint from each interval. Here, you have
x a
x i
i
n
i = +
= +
( )
∆
1 3
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
lim
n
i
i
n
n
i
n
n
i
f x
x
i
n n
n
i
n
→∞ =
→∞ =
→∞ =
∑
∑
=
+ ( )
=
( ) +
∆
1
1
1
1 3 3
3 1 3
n n
∑
532.
lim
sin
n
i
n
n
i
n
→∞ =
( ) ( )
∑
π
π
2
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆x
n
n
= − =
π
π
0
You also want to select a point from each interval. The formula x i = a + (Δ x)i gives you
the right endpoint from each interval. Here, you have
x a
x i
i
n
i
n
i = +
= +
=
( )
∆
0 π
π
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
sin
lim
n
i
i
n
n
i
n
n
f x
x
i
n
n
n
→∞ =
→∞ =
→∞
∑
∑
=
( )
( )
=
∆
1
2
1
π
π
π
( ( ) ( )
=
∑ sin
2
1
πi
n
i
n
533.
lim
n
i
n
i
n
i
n n
→∞ =
+
( ) + +
( )
∑ 1 4
1 4
4
2
1
To begin, you split the interval into n pieces of equal width using the formula
∆x b a
n
= − , where a is the lower limit of integration and b is the upper limit of integration. In this case, you have
∆x
n
n
= − =
5 1 4