Question

Consider the region bounded by f(x) = x^3 + x + 3 and y = 0 over [−1, 2].

a) Find the partition of the given interval into n subintervals of equal length. (Write ∆x, x0, x1, x2, · · · , xk, · · · , xn.)

b) Find f(xk), and setup the Riemann sum ∑k=1 f(xk)∆x.

c) Simplify the Riemann sum using the Power Sum Formulas.

d) Find the area of the region by taking limit as n → ∞ of the simplified Riemann sum.

Answer #1

Consider the function f(x)=4x2-x3 provide
the graph the region bounded by f(x) and the x-axis over the
interval [0,4], then estimate the area of this region using left
reman sum with n=4, 10 and 20 subintervals. you may use the
graphing calculator to facilitate the calculation of the Riemann
sum. use four decimal places in all your calculations and
answers.

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.

Let f(x)=10-2x
a.) Sketch the region R under the graph of f on the interval
[0,5], and find its exact area using geometry.
b.) Use a Riemann sum with five subintervals of equal length
(n=5) to approximate the area of R. Choose the representative
points to be the left endpoints of the subintervals.
c.) Repeat part (b) with ten subintervals of equal length
(n=10).
d.) Compare the approximations obtained in parts (b) and (c)
with the exact area found in...

Let f(x) = x2, and compute
the Riemann sum of f over the interval [5, 7], choosing
the representative points to be the left endpoints of the
subintervals and using the following number of subintervals
(n). (Round your answers to two decimal places.)
(a) two subintervals of equal length (n = 2)
(b) five subintervals of equal length (n = 5)
(c) ten subintervals of equal length (n = 10)
(d) Can you guess at the area of the region...

Find the area of the region bounded by the graph of f(x) = 4x^3 +
4x + 9 and the x axis between x=0 and x=2 using Riemann sums.

6.3 2. Let f(x) = x2, and
compute the Riemann sum of f over the interval [8, 10],
choosing the representative points to be the midpoints of the
subintervals and using the following number of subintervals
(n). (Round your answers to two decimal places.)
a. two subintervals of equal length (n = 2)
___________
b. five subintervals of equal length (n = 5)
__________
c. ten subintervals of equal length (n = 10)
_________
d. Can you guess at the...

Estimate the area of the region bounded between the curve f(x) =
1 x+1 and the horizontal axis over the interval [1, 5] using a
right Riemann sum. Use n = 4 rectangles first, then repeat using n
= 8 rectangles. The exact area under the curve over [1, 5] is ln(3)
≈ 1.0986. Which of your estimates is closer to the true value?

1)| The region bounded by f(x)=−1x^2+5x+14 x=0, and y=0 is
rotated about the y-axis. Find the volume of the solid of
revolution.
Find the exact value; write answer without decimals.
2) Now compute s4, the partial sum consisting of the first 4
terms of ∞∑k=1/7√k5:
s4=
3) Test the series below for convergence using the Ratio
Test.
∞∑n=1 n^2/0.9^n
The limit of the ratio test simplifies to limn→∞|f(n)| where
f(n)=
The limit is:

1) Using the right endpoint with n = 4, approximate the area
of the region bounded by ? = 2?2 + 3, and x axis for x between 1
and 3.
2) Use Riemann sums and the limit to find the area of the
region bounded by ?(?) = 3? − 4 and x-axis between x = 0 and x =
1

Let R denote the region that lies below the graph of y = f(x)
over the interval [a, b] on the x axis. Calculate an underestimate
and an overestimate for the area A of R, based on a division of [a,
b] into n subintervals all with the same length delta(x) = (b -
a)/n.
f(x) = 9 - x2 on [0, 3]; n = 5

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