calc_tutorial_15_3_017

 
Problem Statement

{128}
{15}
round(128/15,2)

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 − 2x2 − y2.

 
Step 1

Question Sequence

Question 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 − 2x2 − y2

x2 + y2 =

Thus the boundary of D is a in the first quadrant.

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 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 D simple.

A.
B.
C.
D.

0 ≤ x ≤

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Write the triple integral as an iterated integral.

A.
B.

A.
B.
Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

Evaluate the inner integral.

A.
B.
Correct.
Incorrect.

 
Step 5

Question Sequence

Question 5

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

A.
B.
C.
D.
Correct.
Incorrect.

 
Step 6

Question Sequence

Question 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.

=

Correct.
Incorrect.