Question

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 part (a) should
approach this number. (Round your answer to three decimal
places.)

square units

Answer #1

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...

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

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

5. A problem to connect the Riemann sum and the Fundamental
Theorem of Calculus:
(a) Evaluate the Riemann sum for f(x) = x 3 + 2 for 0 ≤ x ≤ 3
with five subintervals, taking the sample points to be right
endpoints.
(b) Use the formal definition of a definite integral with right
endpoints to calculate the value of the integral. Z 3 0 (x 3 + 2)
dx.
Note: This is the definition with limn→∞ Xn i=1 f(xi)∆x...

(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

Let R be the region bounded above by f(x) = 3 times the (sqr
root of x) and the x-axis between x = 4 and x = 16. Approximate the
area of R using a midpoint Riemann sum with n = 6 subintervals.
Sketch a graph of R and illustrate how you are approximating it’
area with rectangles. Round your answer to three decimal
places.

1. Evaluate the Riemann sum for
f(x) = 2x − 1, −6 ≤ x ≤ 4,
with five subintervals, taking the sample points to be right
endpoints.
2. sketch a graph
3. Explain.
The Riemann sum represents the net area of the rectangles with
respect to the .....

Let f(x) = e^x. Evaluate a right Riemann sum for the interval
[−1, 1] for n = 4. You should include a picture of the
corresponding rectangles and state if this is an under or over
approximation of the area beneath the graph of f, above the x-axis
and between x = −1 and x = 1. In your solution, you should write
out all terms that go into the Riemann sum.

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