Please help solve the equation using cos(no links)

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n.

Vera_Pavlovna [14]

Split up the integration interval into 4 subintervals:

$\left[0,\dfrac\pi8\right],\left[\dfrac\pi8,\dfrac\pi4\right],\left[\dfrac\pi4,\dfrac{3\pi}8\right],\left[\dfrac{3\pi}8,\dfrac\pi2\right]$

The left and right endpoints of the $i$ -th subinterval, respectively, are

$\ell_i=\dfrac{i-1}4\left(\dfrac\pi2-0\right)=\dfrac{(i-1)\pi}8$

$r_i=\dfrac i4\left(\dfrac\pi2-0\right)=\dfrac{i\pi}8$

for $1\le i\le4$ , and the respective midpoints are

$m_i=\dfrac{\ell_i+r_i}2=\dfrac{(2i-1)\pi}8$

Trapezoidal rule

We approximate the (signed) area under the curve over each subinterval by

$T_i=\dfrac{f(\ell_i)+f(r_i)}2(\ell_i-r_i)$

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4T_i\approx\boxed{3.038078}$

Midpoint rule

We approximate the area for each subinterval by

$M_i=f(m_i)(\ell_i-r_i)$

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4M_i\approx\boxed{2.981137}$

Simpson's rule

We first interpolate the integrand over each subinterval by a quadratic polynomial $p_i(x)$ , where

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

so that

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx$

It so happens that the integral of $p_i(x)$ reduces nicely to the form you're probably more familiar with,

$S_i=\displaystyle\int_{\ell_i}^{r_i}p_i(x)\,\mathrm dx=\frac{r_i-\ell_i}6(f(\ell_i)+4f(m_i)+f(r_i))$

Then the integral is approximately

$\displaystyle\int_0^{\pi/2}\frac3{1+\cos x}\,\mathrm dx\approx\sum_{i=1}^4S_i\approx\boxed{3.000117}$

Compare these to the actual value of the integral, 3. I've included plots of the approximations below.

3 0

3 years ago

Just please help me, im trying to cram multiple things at once (house is crazy) and I just cant seem to concentrate and understa

Anastasy [175]

Y=31.2
X=15.6

I rounded both to the nearest tenth place.

5 0

2 years ago

Read 2 more answers

Sasha packed unit cubes into a prism, as shown. Which statement about the volume of the box is true.

Zina [86]

Answer:

The answer is " The volume is greater than 50 cubic units "

Step-by-step explanation:

In this question, the value of the units and figure is missing that's why its solution can be defined as follows:

Let's the unit value =50 and please find the figure in the attachment file.

The given figure shows its box is not filled with 50 cubes. These were ten empty places where cubes could be placed.

This figure indicates that only the box size will be four units per five units per three units, so they realize it will be in...

$\to (4 \times 5 \times 3) \ units^3 = 60 \ units^3$

That's why the volume value exceeds 50 cubic units.

6 0

2 years ago

What is 2247 divided by 7

UkoKoshka [18]

and 8

Step-by-step explanation:

and and and and and and and and and 8

6 0

3 years ago

Read 2 more answers

How to write Q={1,2,3,5} in set builder form?<br>

blagie [28]

Answer:

{x:x is a natural number less than 6}

Step-by-step explanation:

Set builder is a type of set where we have to represent any value inside a set

by using the variable x, v, b, a, c etc.

Eg A={1,2,3,4}

For eg {a: a is a natural number less than 5}

8 0

2 years ago