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Calc 3 12.3

 

<== Calculus 3
Number Question Answer
See the work
1. Using polar coordinates, evaluate the integral \displaystyle \int \!\! \int_{R} \sin (x^2+y^2) dA where R is the region 4 \leq x^2 + y^2 \leq 81

-4.493 The work
2. By changing to polar coordinates, evaluate the integral
\iint_D (x^2 + y^2)^{9/2} dx dy where D is the disk x^2 + y^2 \le 25.
The value is .
27890559.78 The work
3. Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x^2 + y^2 = 256 and x^2 - 16x + y^2 = 0.

100.531 The work
4. Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant below the line y = 10 and between the circles x^2 + y^2 = 400 and x^2 - 20x + y^2 = 0.

34.243 The work
5. Use the polar coordinates to find the volume of a sphere of radius 6.

904.78 The work
6. Find the volume of the ellipsoid x^2 + y^2 + 5 z^2 = 64.

959.12 The work
7. Find the volume of the solid enclosed by the paraboloids z =
16 \left( x^{2} + y^{2} 
ight) and z = 8 -
16 \left( x^{2} + y^{2} 
ight).

3.142 The work
8. A cylindrical drill with radius 2 is used to bore a hole throught the center of a sphere of radius 5. Find the volume of the ring shaped solid that remains.

403.104 The work
9. A sprinkler distributes water in a circular pattern, supplying water to a depth of e^{-r} feet per hour at a distance of r feet from the sprinkler.

A. What is the total amount of water supplied per hour inside of a circle of radius 14?

{ft}^3 / h

B. What is the total amount of water that goes throught the sprinkler per hour?

{ft}^3 / h

A) 6.2831
B) 6.2831
The work
10. A volcano fills the volume between the graphs z = 0 and \displaystyle z = 
rac{1}{\left( x^{2} + y^{2} 
ight)^{26}}, and outside the cylinder x^{2} + y^{2} = 1. Find the volume of this volcano.

0.1257 The work
11. A. Using polar coordinates, evaluate the improper integral \displaystyle \int \!\! \int_{R^2}
e^{-10 (x^2+y^2)} \ dx \ dy.

B. Use part A to evaluate the improper integral \displaystyle \int_{-\infty}^{\infty}
e^{-10 x^2} \ dx.

A) 0.314159
B) 0.5605
The work