347
Answers
501–600
Answers and Explanations
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 4
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
n
i
i
n
n
i
n
f x
x
i
n
i
n n
→∞ =
→∞ =
=
+
( ) + +
( )
∑
∑
∆
1
2
1
1 4
1 4
4
534.
lim
tan
s ec
n
i
n
n
πi
n
i
n
→∞ =
( ) + ( )
∑
π
π
4
4
4
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
4 0
4
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 = +
= +
( )
∆
0 4
π
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
tan
s ec
n
i
i
n
n
i
n
f x
x
i
n
i
n
n
→∞ =
→∞ =
∑
∑
=
( ) + ( )
∆
1
1
4
4
4
π
π
π
= =
( ) + ( )
→∞ =
∑
lim
tan
s ec
n
i
n
n
i
n
i
n
π
π
π
4
4
4
1
535.
lim
n
i
n
n
i
n
i
n
i
n
→∞ =
( ) +
( ) + +
( ) − +
( ) +
∑
2 3 4 2
4 2
4 2
5
3
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
= − =
6 4 2
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 = +
= +
( )
∆
4 2
Answers
501–600
Answers and Explanations
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 4
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
n
i
i
n
n
i
n
f x
x
i
n
i
n n
→∞ =
→∞ =
=
+
( ) + +
( )
∑
∑
∆
1
2
1
1 4
1 4
4
534.
lim
tan
s ec
n
i
n
n
πi
n
i
n
→∞ =
( ) + ( )
∑
π
π
4
4
4
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
4 0
4
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 = +
= +
( )
∆
0 4
π
Substituting those values into the definition of the definite integral gives you
lim
( )
lim
tan
s ec
n
i
i
n
n
i
n
f x
x
i
n
i
n
n
→∞ =
→∞ =
∑
∑
=
( ) + ( )
∆
1
1
4
4
4
π
π
π
= =
( ) + ( )
→∞ =
∑
lim
tan
s ec
n
i
n
n
i
n
i
n
π
π
π
4
4
4
1
535.
lim
n
i
n
n
i
n
i
n
i
n
→∞ =
( ) +
( ) + +
( ) − +
( ) +
∑
2 3 4 2
4 2
4 2
5
3
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
= − =
6 4 2
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 = +
= +
( )
∆
4 2