Question

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

Answer #1

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

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) =
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...

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...

Approximate the area under the curve y= x2-2x+4
between x=1 and x=4 by using 6 inscribed rectangles.

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?

Approximate the area under the curve over the specified interval
by using the indicated number of subintervals (or rectangles) and
evaluating the function at the right-hand endpoints of the
subintervals.
f(x) = 25 − x2 from x = 1 to x = 3; 4
subintervals

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...

Use finite approximation to estimate the area under the graph of
f(x)=x2 and above the graph of f(x)=0 from
x0=0 to xn =12 using
i) a lower sum with two rectangles of equal width.
ii) a lower sum with four rectangles of equal width.
iii) an upper sum with two rectangles of equal width.
iv) an upper sum with four rectangles of equal width.
The estimated area using a lower sum with two rectangles of
equal width is ______ square...

Approximate the area under the curve y = 12x−4x2 between x = 0
and x = 2 using four rectangles of the same width and the
right-hand endpoint method.

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