Question

Use a graphing calculator *Riemann Sum* (found here) to
find the following Riemann sums.

f(x) =

2/x |

from a = 1 to b = 5

(a) Calculate the Riemann sum for the function for the following
values of *n*: 10, 100, and 1000. Use left, right, and
midpoint rectangles, making a table of the answers, rounded to
three decimal places.

n |
Left | Midpoint | Right |

10 | |||

100 | |||

1000 |

(b) Find the *exact* value of the area under the curve by
evaluating an appropriate definite integral using the Fundamental
Theorem. The values of the Riemann sums from part (a) should
approach this number. (Round your answer to three decimal
places.)

square units

Answer #1

f(x) =
square root x
from a = 4 to b =
9
(a) Calculate the Riemann sum for the function for the following
values of n: 10, 100, and 1000. Use left, right, and
midpoint rectangles, making a table of the answers, rounded to
three decimal places.
n
Left
Midpoint
Right
10
100
1000
(b) Find the exact value of the area under the curve by
evaluating an appropriate definite integral using the Fundamental
Theorem. The values of the Riemann sums from...

For the function, do the following.
f(x) =
1
x
from a = 1 to b =
2.
(a) Approximate the area under the curve from
a to b by calculating a Riemann sum using 10
rectangles. Use the method described in Example 1 on page 351,
rounding to three decimal places.
square units
(b) Find the exact area under the curve from a to
b by evaluating an appropriate definite integral using the
Fundamental Theorem.
square units

f(x) = 1/x
from a = 1 to b =
3.
(a) Approximate the area under the curve from
a to b by calculating a Riemann sum using 10
rectangles. Use the method described in Example 1 on page
351,rounding to three decimal places.
_____________square units
(b) Find the exact area under the curve from a to
b by evaluating an appropriate definite integral using the
Fundamental Theorem.
_____________square units

f(x) = 2x from a =
4 to b = 5
(a) Approximate the area under the curve from
a to b by calculating a Riemann sum using 5
rectangles. Use the method described in Example 1 on page 351,
rounding to three decimal places.
square units
(b) Find the exact area under the curve from a to
b by evaluating an appropriate definite integral using the
Fundamental Theorem.
square units

(a) Find the Riemann sum for
f(x) = 3
sin(x), 0 ≤ x ≤
3π/2,
with six terms, taking the sample points to be right endpoints.
(Round your answers to six decimal places.)
R6 =
(b) Repeat part (a) with midpoints as the sample points.
M6 =
Express the limit as a definite integral on the given
interval.
lim n → ∞
n
7xi* +
(xi*)2
Δx, [3, 8]
i = 1
8
dx
3

Instructions: Approximate the following definite integrals using
the indicated Riemann sums.
1. Z 9 1 x 1 + x dx using a left-hand Riemann sum L4 with n = 4
subintervals.
2. Z 3 0 x 2 dx using a midpont Riemann sum M3 using n = 3
subintervals.
3. Z 3 1 f(x) dx using a right-hand Riemann Sum R4, with n = 4
subintervals

Use upper and lower sums (left and right Riemann sums) to
approximate the area
of the region below y =sqrt(8x) using 4 subintervals of each width.
Round to three
decimal places.

With a programmable calculator (or a computer), it is possible
to evaluate the expressions for the sums of areas of approximating
rectangles, even for large values of n, using looping. (On
a TI use the Is> command or a For-EndFor loop, on a Casio use
Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of
the areas of approximating rectangles using equal subintervals and
right end points for n = 10, 30, 50, and 100....

In the Midpoint Rule for triple integrals we
use a triple Riemann sum to approximate a triple integral over a
box B, where
f(x, y, z)
is evaluated at the center
(xi, yj, zk)
of the box
Bijk.
Use the Midpoint Rule to estimate the value of the integral.
Divide B into eight sub-boxes of equal size. (Round your
answer to three decimal places.)
cos(xyz) dV, where B = {(x, y, z) | 0 ≤ x ≤ 2, 0 ≤...

In the Midpoint Rule for triple integrals we use a triple
Riemann sum to approximate a triple integral over a box B, where
f(x, y, z) is evaluated at the center (xi, yj, zk) of the box Bijk.
Use the Midpoint Rule to estimate the value of the integral. Divide
B into eight sub-boxes of equal size. (Round your answer to three
decimal places.) cos(xyz) dV, where B = {(x, y, z) | 0 ≤ x ≤ 5, 0 ≤...

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