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myrzilka [38]
3 years ago
12

Use Simpson's Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare y

our answer with the value of the integral produced by a calculator. (Round your answers to the nearest whole number.) y = 1 5 x5, 0 ≤ x ≤ 5
Mathematics
1 answer:
DiKsa [7]3 years ago
4 0

The area of the surface is given exactly by the integral,

\displaystyle\pi\int_0^5\sqrt{1+(y'(x))^2}\,\mathrm dx

We have

y(x)=\dfrac15x^5\implies y'(x)=x^4

so the area is

\displaystyle\pi\int_0^5\sqrt{1+x^8}\,\mathrm dx

We split up the domain of integration into 10 subintervals,

[0, 1/2], [1/2, 1], [1, 3/2], ..., [4, 9/2], [9/2, 5]

where the left and right endpoints for the i-th subinterval are, respectively,

\ell_i=\dfrac{5-0}{10}(i-1)=\dfrac{i-1}2

r_i=\dfrac{5-0}{10}i=\dfrac i2

with midpoint

m_i=\dfrac{\ell_i+r_i}2=\dfrac{2i-1}4

with 1\le i\le10.

Over each subinterval, we interpolate f(x)=\sqrt{1+x^8} with the quadratic polynomial,

p_i(x)=f(\ell_i)\dfrac{(x-m_i)(x-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+f(m_i)\dfrac{(x-\ell_i)(x-r_i)}{(m_i-\ell_i)(m_i-r_i)}+f(r_i)\dfrac{(x-\ell_i)(x-m_i)}{(r_i-\ell_i)(r_i-m_i)}

Then

\displaystyle\int_0^5f(x)\,\mathrm dx\approx\sum_{i=1}^{10}\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx

It turns out that the latter integral reduces significantly to

\displaystyle\int_0^5f(x)\,\mathrm dx\approx\frac56\left(f(0)+4f\left(\frac{0+5}2\right)+f(5)\right)=\frac56\left(1+\sqrt{390,626}+\dfrac{\sqrt{390,881}}4\right)

which is about 651.918, so that the area is approximately 651.918\pi\approx\boxed{2048}.

Compare this to actual value of the integral, which is closer to 1967.

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60.April would like to invest $200 in the bank for one year. Three banks all have a nominal APR of 1.5%, but compound the intere
mojhsa [17]

Answer:

$203.02

Step-by-step explanation:

Since the bank in question compounds interest continuously, the following equation should be used to determine the final balance (B) in April's account:

B= 200*e^{i*t}

Where "e" is a mathematical constant approximated as 2.7183, "i" is the interest rate (1.5%) and "t" is the investment time in years (1):

B= 200*e^{0.015*1}

B= 203.02

April's balance would be $203.02 after one year.

5 0
3 years ago
Select the function that represents a geometric sequence.
jeka57 [31]

Answer:

A.

A(n) = P(1 + i)^n-1, where n is a positive integer

6 0
2 years ago
Read 2 more answers
5. Find the general solution to y'''-y''+4y'-4y = 0
CaHeK987 [17]

For any equation,

a_ny^(n)+\dots+a_1y'+a_0y=0

assume solution of a form, e^{yt}

Which leads to,

(e^{yt})'''-(e^{yt})''+4(e^{yt})'-4e^{yt}=0

Simplify to,

e^{yt}(y^3-y^2+4y-4)=0

Then find solutions,

\underline{y_1=1}, \underline{y_2=2i}, \underline{y_3=-2i}

For non repeated real root y, we have a form of,

y_1=c_1e^t

Following up,

For two non repeated complex roots y_2\neq y_3 where,

y_2=a+bi

and,

y_3=a-bi

the general solution has a form of,

y=e^{at}(c_2\cos(bt)+c_3\sin(bt))

Or in this case,

y=e^0(c_2\cos(2t)+c_3\sin(2t))

Now we just refine and get,

\boxed{y=c_1e^t+c_2\cos(2t)+c_3\sin(2t)}

Hope this helps.

r3t40

5 0
3 years ago
A 1/4 lb package of sunflower seeds cost 79cents. An 8 ounces package cost 1.59. Which package represents the lower cost per oun
arsen [322]
The first package is 4 oz (1/4lb) at 79cents = 19.75 cents/oz
Second Package is 8 oz at 1.59 = 19.875 cents / oz
So the first package is cheaper per ounce.
6 0
3 years ago
Wiebe Trucking. Inc. is planning a new warehouse to serve the west. Denver, Santa Fe, and Salt Lake City are under consideration
murzikaleks [220]

The total cost curve shows the cost of total shipment from the different

cities.

  • a. Please find attached the graph of the total cost to quantity of shipment created with MS Excel.

  • b. The city that provides the lowest overall cost is; <u>Salt Lake City</u>.

Reasons:

The given parameters are;

The number of shipment per = From 550,000 to 600,000 per year

The given table is presented as follows;

\begin{tabular}{|l|c|c|c|}Location&Annual Fixed Costs&Variable Cost \\Denver&\$5,000,000&\$4.65 \\Santa Fe&\$4,200,000&\$6.25\\Salt Lake City&\$3,500,000&\$7.25\end{array}\right]

a. Required:

The plot total cost curve for the locations on a single graph.

Solution:

  • Please find attached the graph of the total cost curves created with MS Excel

b. Required:

The city that provides the lowest overall cost.

Solution:

The two cities with the lowest overall costs are Denver and Salt Lake City.

  • From the total cost curve, the area under the curves are;

Area under the curve for Denver;

(7557500 + 7790000) ÷ 2 × 50,000 = 383687500000

Area under the curve for Salt Lake City

(7487500 + 7850000) ÷ 2 × 50,000 = 383437500000

Therefore;

  • <u>Salt Lake City provides the lowest overall costs</u>.

Learn more about total cost curves here:

brainly.com/question/4888738

4 0
2 years ago
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