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I need a fully explanation and calculation for each questions with their correct answers. I have uploaded the questions documantes below.

V1

MAT 2311: Calculus I

Quiz #9: Sections 5.6-6.2

Directions: Do on a separate sheet. To receive credit, you must show a clear explanation and sufficient

work to support the answers you give. Make sure your work is neat and orderly, and in the correct

notation. All answers should be exact and simplified unless otherwise indicated. Circle your final

answer.

Note: Correct sketches are the key to choosing dx or dy, and using the correct measurements.

Follow the directions for each question. In particular, sketch and label one “piece” – rectangles for

area, disks/washer/shell for volumes.

1. Sketch and shade the region bounded by the graphs of the equations and use integrals to find the area.

Also sketch 1 representative rectangle.

3

a.

=

y x=

, y 4x

b. =

, x 0,=

y ln x=

y 0,=

y 1

2. Sketch the region R bounded by the graphs of the equations, and find the volume of the solid

generated by revolving R about the indicated axis.

Sketch the shaded region, the solid, and a typical disk/washer/shell..

1

a. =

y

,=

y 1,=

y 2,=

x 0; about the y-axis

x

b. y = x 2 , y = 1 ; about the line y = −1 .

c. =

y sin ( x 2 ) ,=

y 0,=

x 0,=

x

π ; about the line x = 0.

(For part c, use a graphing calculator to obtain a sketch of the graph).

3. Given the region bounded by x =

2 y − y 2 and x =

0, set up an integral and find the volume of the

solid:

y

3

a) When the region is rotated about the y-axis

2

b) When the region is rotated about the x-axis.

Sketch the shaded region, the solid, and a typical

disk/washer/shell for both questions.

=

x 2 y − y2

1

x

-1

0

1

2

-1

4. (Extra Credit)

The circumference of a certain tree at different heights above the ground is given in the following

table:

height (ft)

0

20 40 60 80 100 120

circumference(ft)

26 22 19 14

6

3

1

Assume all horizontal cross sections are circular. Make a sketch and set up an integral to find the

volume for the tree. Then use the Midpoint Rule to approximate the volume of the tree (to the nearest

whole number).

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