Chapter 1. calc_tutorial_15_3_017

1.1 Problem Statement

{128}
{15}
round($a/$b,2)

Integrate f(x, y, z) = x over the region in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) above z = y2 and below z = 8 − 2x2y2.

1.2 Step 1

Question Sequence

Question 1.1

Find the boundary of D in the xy-plane. The upper and lower surfaces intersect where they have the same z-values. Find the intersection of the two surfaces in terms of x and y ≥ 0.

y2 = 8 − 2x2y2

x2 + y2 = h4XZagboIgc=

Thus the boundary of D is a 3Fxr8xmp24lukJ+1AGBVkCFMSN3GWMbf8e5lWEhHnCmTFEDt in the first quadrant.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

The region D can be expressed as either vertically or horizontally simple, depending on which variable we express the boundary in.

Solve the boundary, x2 + y2 = 4, for y in terms of x, thus making DM2jxtJGVDpVt/Kx8Ux1JJ4ofYkdaWFI9ZSKzvQ== simple.

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

0 ≤ xXvVM00l89Is=

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Write the triple integral as an iterated integral.

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
Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Evaluate the inner integral.

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
Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

At this point, we will choose to write the integrand as 8x − 2x3 − 2xy2 = 2x(4 − x2) − 2xy2. Integrate the middle integral.

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
Correct.
Incorrect.

1.7 Step 6

Question Sequence

Question 1.6

Evaluate f(x, y, z) = x over the defined region by integrating the last of the iterated integral. Round your answer to two decimal places.

= SFgqQUkJGdg=

Correct.
Incorrect.