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Calc 3 12.4

 

<== Calculus 3
Number Question Answer
See the work
1. Electric charge is distributed over the disk
x^2 + y^2 \leq 7 so that the charge density at (x,y) is \sigma(x,y) = 17 + x^2 +
 y^2 coulombs per square meter.
Find the total charge on the disk.

450.819 The work
2. Find the mass of the rectangular region 0 \leq x \leq 2 with density function 
ho \left( x, y 
ight) = 1 - y.

1 The work
3. Find the mass of the triangular region with vertices (0, 0), (1, 0), and (0, 4), with density function 
ho \left( x, y 
ight) = x^2 + y^2.

17/3 The work
4. A lamina occupies the part of the rectangle 0 &le x &le 6, 0 &le y &le 4 and the density at each point is given by the function 
ho(x,y) = 6 x + 5 y + 3.

A. What is the total mass?

B. Where is the center of mass? ( , )

A) 744

B) ( 3.581, 2.2151)
The work
5. A lamina occupies the region inside the circle x^2 + y^2 = 6 y but outside the circle x^2 + y^2 = 9. The density at each point is inversely proportional to its distance from the orgin.

Where is the center of mass?
( , )

( 0, 3.794) The work
6. A lamina occupies the part of the disk x^2 + y^2 \leq 25 in the first quadrant and the density at each point is given by the function 
ho(x,y) = (x^2+y^2).

A. What is the total mass?

B. What is the moment about the x-axis?

C. What is the moment about the y-axis?
D. Where is the center of mass? (, )

E. What is the moment of inertia about the origin?
A. Total mass: 245.437
B. 625
C. 625
D. Center of mass: ( 2.546, 2.546)
E. 4090.62
The work
7. You are getting married and your dearest relative has baked you a cake which fills the volume between the two planes, z = 0 and z = 1x + 7y + c, and inside the cylinder x^2 + y^2 = 1. You are to cut it in half by making two vertical slices from the center outward. Suppose one of the slices is at &theta = 0 and the other is at &theta = &psi .

What is the limit, \displaystyle \lim_{c 
ightarrow \infty} \psi?

&pi The work
8. A lamp has two bulbs, each of a type with an average lifetime of 3 hours. The probability density function for the lifetime of a bulb is f(t) = rac{1}{3}
e^{-t/3}, t \leq 0.
What is the probability that both of the bulbs will fail within 2 hours?

0.2368 The work