Mathematics, 03.06.2020 19:02 sosahaded

A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write

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Answer from: lin43

Writing the integral in polar coordinates is less taxing:

$\displaystyle\iint_Rf(x,y)\,\mathrm dA=\iiint_Rrf(r\cos\theta,r\sin\theta)\,\mathrm dr\,\mathrm d\theta$

The region R is captured by the set of points (in polar, of course)

$R=\left\{(r,\theta)\mid1\le r\le3\land\dfrac{5\pi}4\le\theta\le2\pi\right\}$

so that the integral is

$\displaystyle\int_{5\pi/4}^{2\pi}\int_1^3rf(r\cos\theta,r\sin\theta)\,\mathrm dr\,\mathrm d\theta$

In rectangular coordinates, we have to split up the region so that

$R=R_1\cup R_2\cup R_3$

where

$R_1=\left\{(x,y)\mid-\dfrac3{\sqrt2}\le x\le-\dfrac1{\sqrt2}\land-\sqrt{9-x^2}\le y\le x\right\}$

$R_2=\left\{(x,y)\mid-\dfrac1{\sqrt2}\le x\le1\land-\sqrt{9-x^2}\le y\le -\sqrt{1-x^2}\right\}$

$R_3=\left\{(x,y)\mid1\le x\le3\land-\sqrt{9-x^2}\le y\le0\right\}$

and the integral is split into 3 over these regions,

$\displaystyle\iint_Rf(x,y)\,\mathrm dA=\sum_{n=1}^3\iint_{R_n}f(x,y)\,\mathrm dx\,\mathrm dy$

A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write

Answer from: Quest

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer: (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

Answer from: Quest

kjgfcvbjkl

step-by-step explanation:

sdfghjkl; vbnm,

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A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write

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