What is 130° in radians?

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

Solve the inequality and express your answer in interval notation. X^2-6x+7<0

kakasveta [241]

Answer:

15

Step-by-step explanation:

2×6=8

8+7 =15

2+7 =9

9

8 0

3 years ago

Marla bought 7 boxes. A week later, half of all her boxes were destroyed in a fire. There are now only 22 boxes left. How many d

Alchen [17]

She should have started with 44 boxes

3 0

3 years ago

Read 2 more answers

In a certain pond, 50 fish were caught, tagged, and returned to the pond. A few days later, 50 fish were caught again, of which

mario62 [17]

2 out of 50 were tagged.

Divide 2 by 50:

2/50 = 0.4 ( This is 4% of the fish were tagged).

Now divide the number of fish caught by the percentage that were tagged:

50 / 0.04 = 1250

The number of fish in the pond is C. 1250

6 0

2 years ago

How do you find the missing number of a perfect square trinomial?<br><br> 25x^2 - 60x + ___

Harrizon [31]

Answer:

36

Step-by-step explanation:

Compare what you have to the square ...

(a +b)^2 = a^2 +2ab +b^2

Your "a" is √(25x^2) = 5x

Your "2ab" is -60x. Since you know "a", you can find "b".

2ab = -60x

2(5x)b = -60x . . . . . . . substitute for "a"

b = -60x/(10x) = -6

Then the missing term is b^2 = (-6)^2 = 36.

Your trinomial is ...

25x^2 -60x +<u>36</u>

4 0

2 years ago