Question

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.

Answer #1

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...

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...

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.

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?

Use the midpoint rule with 4 rectangles to approximate the area
of the region bounded above by y=sinx, below by the ?x-axis, on
the left by x=0, and on the right by ?=?

For the function f(x) = x, estimate the area of the region
between the graph and the horizontal axis over the interval 0≤x≤4
using a .
a. Riemann Left Sum with eight left rectangles.
b. Riemann Right Sum with eight right rectangles.
c. A good estimate of the area.

Let R be the region bounded by the curves y = x, y = x+ 2, x =
0, and x = 4. Find the volume of the solid generated when R is
revolved about the x-axis. In addition, include a carefully labeled
sketch as well as a typical approximating disk/washer.

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

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.

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 .....

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