339
Answers
501–600
Answers and Explanations
calculate the heights of the rectangles gives you the values x = 0, x = 1
2
, x = 1, and x = 3
2
.
The height of each rectangle is f
(x). To approximate the area under the curve, multiply
the width of each rectangle by f
(x) and add the areas:
1
2
0 1
2
1
2
1
2
1 1
2
3
2
1
2
2 0
1
2
2 1
2
1
2
2
f
f
f
f
( )
( ) ( ) ( )
( )
+
+
+
=
+
+
+
+ 2 2
2 1
1
2
2 3
2
23
4
2
2
+
+
+
=
( )
( )
520.
10.43
Because the given function has values that are strictly greater than or equal to zero on
the given interval, you can interpret the Riemann sum as approximating the area that’s
underneath the curve and bounded below by the x-axis.
You want to use five rectangles of equal width to estimate the area under f x
x x
( ) =
+
3
over the interval 1 ≤ x ≤ 4. Begin by dividing the length of the interval by 5 to find the
width of each rectangle:
∆
− =
x = 4 1
5
3
5
Then divide the interval [1, 4] into 5 equal pieces, each with a width of 3
5
, to get the
intervals 1 8
5
,
, 8
5
11
5
,
, 11
5
14
5
,
, 14
5
17
5
,
, and 17
5
4
,
. Using the left endpoint of each
interval to calculate the heights of the rectangles gives you the values x = 1, x = 8
5
,
x = 11
5
, x = 14
5
, and x = 17
5
. The height of each rectangle is f
(x). To approximate the area
under the curve, multiply the width of each rectangle by f
(x) and add the areas:
3
5
1 3
5
8
5
3
5
11
5
3
5
14
5
3
5
17
5
3
5
1 1 3
5
8
5
3
3
f
f
f
f
f
( ) +
( ) + ( ) + ( ) ( )
+
=
+ +
+ +
+
+
+
+
+
+
8
5
3
5
11
5
11
5
3
5
14
5
14
5
3
5
17
5
17
5
3
3
3
≈ 10 43
.
521.
22.66
Because the given function has values that are strictly greater than or equal to zero on
the given interval, you can interpret the Riemann sum as approximating the area that’s
underneath the curve and bounded below by the x-axis.
You want to use seven rectangles of equal width to estimate the area under f
(x) =
4 ln x + 2x over the interval 1 ≤ x ≤ 4. Begin by dividing the length of the interval by 7 to
find the width of each rectangle:
∆ = − =
x 4 1
7
3
7
Answers
501–600
Answers and Explanations
calculate the heights of the rectangles gives you the values x = 0, x = 1
2
, x = 1, and x = 3
2
.
The height of each rectangle is f
(x). To approximate the area under the curve, multiply
the width of each rectangle by f
(x) and add the areas:
1
2
0 1
2
1
2
1
2
1 1
2
3
2
1
2
2 0
1
2
2 1
2
1
2
2
f
f
f
f
( )
( ) ( ) ( )
( )
+
+
+
=
+
+
+
+ 2 2
2 1
1
2
2 3
2
23
4
2
2
+
+
+
=
( )
( )
520.
10.43
Because the given function has values that are strictly greater than or equal to zero on
the given interval, you can interpret the Riemann sum as approximating the area that’s
underneath the curve and bounded below by the x-axis.
You want to use five rectangles of equal width to estimate the area under f x
x x
( ) =
+
3
over the interval 1 ≤ x ≤ 4. Begin by dividing the length of the interval by 5 to find the
width of each rectangle:
∆
− =
x = 4 1
5
3
5
Then divide the interval [1, 4] into 5 equal pieces, each with a width of 3
5
, to get the
intervals 1 8
5
,
, 8
5
11
5
,
, 11
5
14
5
,
, 14
5
17
5
,
, and 17
5
4
,
. Using the left endpoint of each
interval to calculate the heights of the rectangles gives you the values x = 1, x = 8
5
,
x = 11
5
, x = 14
5
, and x = 17
5
. The height of each rectangle is f
(x). To approximate the area
under the curve, multiply the width of each rectangle by f
(x) and add the areas:
3
5
1 3
5
8
5
3
5
11
5
3
5
14
5
3
5
17
5
3
5
1 1 3
5
8
5
3
3
f
f
f
f
f
( ) +
( ) + ( ) + ( ) ( )
+
=
+ +
+ +
+
+
+
+
+
+
8
5
3
5
11
5
11
5
3
5
14
5
14
5
3
5
17
5
17
5
3
3
3
≈ 10 43
.
521.
22.66
Because the given function has values that are strictly greater than or equal to zero on
the given interval, you can interpret the Riemann sum as approximating the area that’s
underneath the curve and bounded below by the x-axis.
You want to use seven rectangles of equal width to estimate the area under f
(x) =
4 ln x + 2x over the interval 1 ≤ x ≤ 4. Begin by dividing the length of the interval by 7 to
find the width of each rectangle:
∆ = − =
x 4 1
7
3
7