Set up the triple integral for the volume of the solid that is the common interior below the sphere x2+ y2+ z2= 8 and above the paraboloid z= 1/2 (x2+ y2)

Step-by-step explanation:

The common region enclosed above the paraboloid and below the sphere is the solid.

Let's find the limit bounds for .

The region is bounded above by the sphere .

Isolate from the equation above.

On the other hand, the region is bounded below by the paraboloid  

Therefore, we obtain

Now, we need to find the intersection of the sphere and the paraboloid. To do so, we need to solve the following system of equations

.

Substitute  the second equation in the first equation. We obtain

Hence, the paraboloid and the sphere intersect when .

Substituting for in the equation above gives

which is a circle with a radius .

Now, we can find the bond for and .

For , we obtain

.

For , we have

.

Therefore, the needed triple integral is

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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anser is a

step-by-step explanation:

negitive slope