If 40 is increased to 56, what is the percent of change? <br> a80% <br> b29% <br> c40% <br> d71%

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

DiKsa [7]

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.

4 0

3 years ago

Given the following similar triangles, find x.

Vsevolod [243]

I’ll help if u attach the pic of the question:)

3 0

2 years ago

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A copy machine makes 32 copies per minute. How many copies does it make in 4 minutes and 15 seconds?

ki77a [65]

Answer:

136

Step-by-step explanation:

32*4=128

15=1/4 of a minute

32/4=8

128+8=136

6 0

3 years ago

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Matt uses a compass and straightedge to construct parallel lines. Annie uses technology. In your own words describe how Matts co

ra1l [238]

Thank you for posting your question here at brainly. I hope the answer will help you. Feel free to ask more questions here.

Matt's steps will be more complicated since Annie is using a program to do the constructions hers would probably be precise and easy.

5 0

2 years ago

Zachary invests $400 into an account that earns 3.5% simple interest for 5 years. He does not make any other deposits or withdra

Zinaida [17]

Answer:

Zachary invests $470 into the account that earns 5% simple interest. At the end of 2 years, the account balance is $517

Step-by-step explanation:

I took the test

3 0

2 years ago

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If 40 is increased to 56, what is the percent of change? a80% b29% c40% d71%