Question

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 (

c) Use the Fundamental Theorem of Calculus to check your answer in part (b).

Answer #1

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 .....

(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

Evaluate the Riemann sum for f ( x ) = ln ( x ) − 0.9 over the
interval [ 1 , 5 ] using eight subintervals, taking the sample
points to be right endpoints. R 8 = step by step and answer
please..

The Fundamental Theorem of Calculus allows the calculation of
definite integrals given an integration interval. Not only for this
reason, this Theorem is very important for another factor.
Considering this information, it can be said that the
Fundamental Theorem of Calculus is relevant to Calculus, also
because:
A) it is the only theorem that involves integrals.
B) it makes the use of derivatives unnecessary.
C) it refutes the Riemann integral.
D) it performs the connection of the Integral Calculus with...

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

Evaluate the Riemann sum for
f(x)=0.4x−1.8sin(2x)f(x)=0.4x-1.8sin(2x) over the interval
[0,2][0,2] using four subintervals, taking the sample points to be
right endpoints.
R4=
step by step with answer please

Express the given integral as the limit of a Riemann sum but do
not evaluate: the integral from 0 to 3 of the quantity x cubed
minus 6 times x, dx.

1. Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
2. Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
y =
4
u3
1 + u2
du
2 − 3x
3. Evaluate the integral.
4. Evaluate the integral.
5. Evaluate the integral.
6. Find the derivative of the function.

A particle is moving with the given data. Find the position of
the particle. a(t) = t2 − 5 t + 4, s'(0) = 0, s(1) = 1 s(t) =
Consider the function f(x) = x^2 - 2 x. Sketch the graph of f(x)
and divide the closed interval [-2,4] into 3 equal subintervals (To
get full credit, you must sketch the graph and corresponding
rectangles in your submitted work). a. Sketch the corresponding
rectangles by using Right endpoints in...

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...

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