Question

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?

Answer #1

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.

Estimate the area under the graph of f ( x ) = 1 x + 1 over the
interval [ 3 , 5 ] using two hundred approximating rectangles and
right endpoints
R n =
Repeat the approximation using left endpoints
L n =

Estimate the area under the graph of f(x)=1/(x+3) over the
interval [−2,1] using ten approximating rectangles and
right endpoints. a=-2,b=1,n=10
Rn=?
Repeat the approximation using left endpoints.
Ln?
Accurate to 4 places.

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.

30) Estimate the area under the graph of f(x)= 1/x+4 over the
interval [1,3] using five approximating rectangles and
right endpoints.
Rn=
Repeat the approximation using left endpoints.
Ln=

Estimate the area under the graph of f(x)=1/(x+2) over the
interval [0,3]using eight approximating rectangles and
right endpoints.
Rn=
Repeat the approximation using left endpoints.
Ln=

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...

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 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.

Calculate the area, in square units, of the region bounded by
the line x=2 on the left, the curve f(x)=ln(x-6)+1 on the right,
the line y=3 above, and the x-axis below. Give an exact answer, in
terms of e.

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