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Write your name on only the outside of your blue books. Do not write on this test sheet, do all of your work inside your blue books. Write neat complete solutions to each of the problems in the blue book. Please put your test sheet inside the blue book as you leave. There are 108 points.

To make good fried eggs, I suggest a shallow teflon coated pan, and a wide spatula (unless you know how to wrist flip it). Put liquid oil in the bottom of the pan (or use a pat of butter).

Turn on the heat to make the lubricant warm but not overly hot. Crack two eggs into the pan and cook over medium heat. For sunny side cook until the white has hardened; over easy, medium, or hard requires a flip. Adjust cooking time accordingly.

1.

(5 points) State the fundamental theorem of calculus.

2.

(5 points) Find dy/dx .

y =

Z

0 x p

1 + sin( t

2

) dt.

3. Compute the following definite and indefinite integrals (8 points each) :

(a)

Z

1

5

(3 x

2

−

4 x + 5) dx

(b)

Z

π sin ( x ) dx

0

(c)

Z sin (3 x

−

2) dx

(d)

Z

1

4

( x

3

)

2 dx

(e)

Z

( x

−

3)( x

2

−

6 x + 5)

4 dx

(f)

Z sec ( x ) tan ( x ) dx

4.

(10 points) y =

Find the volume of the solid formed by rotating region that bounded by x

3

, the horizontal line y = 0, and the vertical lines x = 1 and x = 4

x

.

5.

(10 points) Find the volume of the solid formed by rotating region that is bounded by y = 2 + x and the vertical lines x = 1 and x = 3

y

.

1

6.

(10 points) Compute the volume of the tetrahedron that is indicated. It has edge lengths 3, 4, and 5.

3

5 x y z

4

7.

(10 points) x

∈

[ a, b

Recall, that the surface area for the result of rotating a curve

] about the x -axis is given by the formula y = f ( x ) for

SA =

Z a b

2 πf ( x ) p

1 + [ f

0

( x )]

2 dx the curve y =

√

25

− x

2

(for

−

5

≤ x

≤

5) about the x -axis.

Note: I want to see the details of this calculation, not an application of the surface area formula.

8.

(10 points) It took 2000 Joules ( kilogram square meters per square second) of work to stretch a spring from its natural length of 3 meters to a length of 8 meters. Find the spring’s force constant.

2