Chapter 1. calc_tutorial_15_4_017

1.1 Problem Statement

{2,4,6,8}
round($a/8,2)
round(4*$b,2)

Calculate the integral over the given region by changing to polar coordinates.

f(x, y) = |$axy|; x2 + y2 ≤ 1

1.2 Step 1

Question Sequence

Question 1.1

Express region D = {(x, y) | x2 + y2 ≤ 1} in polar coordinates.

0 ≤ r ≤ 0VV1JcqyBrI=

0 ≤ θ ≤ XvVM00l89Is=π

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Express the double integral in polar coordinates.

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
Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Evaluate the inner integral and use the trigonometric identity sin 2θ = 2 cos θ sin θ. Round your answer to two decimal places.

= iSba6t70dtA=

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

In order to evaluate , we must find where sin 2θ is positive and negative in the interval [0, 2π].

By graphing sin 2θ, state the values θ where |sin 2θ| = sin 2θ. That is, find where sin 2θ is positive.

0 ≤ θ ≤

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

and

π ≤ θ ≤

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

State the values θ where |sin 2θ| = −sin 2θ. That is, find where sin 2θ is negative.

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

≤ θ ≤ π

and

Ve/ShTMej63PHN0n5/boca/+8FUpCpLpXeIKw1fcdTn7VblzQ78abwB2TDtQej1YyfVrduLVztuyi+Zh1tut4VFBg7G3Q5pXIjONVxIH4hMN8J36OP8R6KCBG981Obdahrvu/fL5Ju8jB2Rn5Gsz1rj12rBkAirREw7IrASOENtiBlSQLLDXoi6jl5++JvrRbPyJoVTSxWhlWrOxGsy9OuPoI51z6+vvqsh3GOeyABQefeUoSvzA4JtArtdFGwEbAHDgqDdfpzAY5k07vGBkNowSeOiIbIRVfSf5EDAF2I7t6IAl8czO3/RKTN3EpnFM91TodgK0i6R5XoaM4nQ+t2RLebRcVsQI5RJgVXi6ABaaT9WGHDARyxteJ+YjHUuGUXrpNu4l0FysN1+zVrE1X2j8n4KgRCRMeS9+/jR/9Zv6mGanqkwm5rsKh58x+tOpsVtTzM5U9cOYmmsxAJ0nD8QyQNaR5vStlCzeAx3Ic9wltEHwmOyfAsoFe8aEEU4K

≤ θ ≤ 2π

Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

Using the intervals found above, express as a sum/difference of positive integrals.

= brSjF5lOKMQ=9DxysKU6P+4=brSjF5lOKMQ=9DxysKU6P+4=

Correct.
Incorrect.

1.7 Step 6

Question Sequence

Question 1.6

Evaluate .

= h4XZagboIgc=

Correct.
Incorrect.

1.8 Step 7

Question Sequence

Question 1.7

Evaluate using polar coordinates. Round your answer to two decimal places.

= SFgqQUkJGdg=

Correct.
Incorrect.