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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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Answer:

<em>4 blue marbles</em>

<u>Skills needed: Number Theory, Fractions</u>

Step-by-step explanation:

1) The problem tells us that the jar contains marbles that are either red, green, or blue (No Other Color).

It also mentions that \frac{2}{5} (\frac{4}{10}) of the marbles are red

and that \frac{1}{10} of the marbles are blue

- The given statement is that 20 marbles are green, and we need to find out the number of blue marbles.

2) We need to first find out the total number of marbles, and then multiply that by \frac{1}{10} to get the # of blue marbles.

---> Given \frac{4}{10} are red and \frac{1}{10} are blue, 1-\frac{1}{10}-\frac{4}{10} of the marbles are green (since fractions add up to <u>1 whole</u><u>)</u>

<u />1 - \frac{1}{10} - \frac{4}{10} = \frac{9}{10} - \frac{4}{10}=\frac{5}{10}<u />

<u />\frac{5}{10} \text{ is also } \frac{1}{2}<u />

Half the marbles are green.

3) Let's make the total marbles the variable: m

\frac{1}{2} * m=20 ---> Since half of (of signifies multiplication) the total marbles are green.

Multiply both sides by 2 to isolate: m = 20*2, m=40

Total number of marbles is 40.

4) After that, we multiply this by \frac{1}{10} since one-tenth of the total is blue.

---> \frac{1}{10}*m = \frac{1}{10}*40=4

4 blue marbles is the answer.

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