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TEACHER

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Calculus Calculus - pre

alculate the area of of thealculate area Given theGiven equations linesof or or circles inequalities of set a thata bound triangular, rectangular, trapezoidal,or circular region, the calculate volume area and/or surface the of region formed revolving the by region about a horizontalor vertical line. (200_PC.AV_B03) Pre Skills/Objectives C triangle, a rectangle, ortrapezoid, circle, these of composite by formed figures linear equations, orlinear inequalities, conic equations the and/or determine equations the of lines and that circles bound the figure. (200_PC.AV_B02)

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Areas and Volumes and Areas

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Being able to visualize solids is a valuable tool for calculus able tool to visualize for solidsBeing is a valuable calculating conceptual conceptual tape a triangle onto a stick or dowel and then rotate the triangle horizontally triangle horizontally triangle and then the tapeand a onto rotate dowel or a stick Shade the region to be to be r the region Shade t of the (mirror image) reflection Draw with points reflections and their significant ellipses. Connect Draw the boundaries. the Draw Students construct and use manipulatives. and use Students construct situation. knowledge to a new Students apply Volumes and Areas Volumes and Areas Students engage in independent practice. in independent Students engage drawing a sketch of solid revolution sketch a a drawing

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TEACHER

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TEACHER

Volumesof Revolution

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TEACHER

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TEACHER

viii

Volumesof Revolution

– Answers

Mathematics

Volumes of Revolution

1. Sketch the region bounded by the lines y = 3, y = 1, x = 1, x = 6.

c) 5 4 3 2 1

-2 -1 -1 1 2 3 4 5 -2 -3 -4 -5

d) Determine the perimeter of the region.

e) Determine the area of the region.

f) Draw a picture of the region being revolved about the x-axis.

g) Describe the geometric solid formed by revolving the region about the x-axis.

h) Determine the volume of the geometric solid.

3 2. Sketch the region bounded by the lines y = x – 3, y = 0, x = 0. 4

a) 5 4 3 2 1

-2 -1 -1 1 2 3 4 5 -2 -3 -4 -5

b) Determine the perimeter of the region.

c) Determine the area of the region.

d) Draw a picture of the region being revolved about the x-axis.

e) What geometric figure is formed by revolving the region about the x-axis?

f) Determine the volume of the geometric solid.

g) Determine the surface area of the geometric solid.

h) If the region were revolved about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer. Compare the surface areas.

i) Name another region that could be revolved about the x-axis to create exactly the same geometric solid.

3. Sketch the region bounded by the curve y = 4  x2 and the line y = 0.

a) 5 4 3 2 1

-2 -1 -1 1 2 3 4 5 -2 -3 -4 -5

b) Determine the perimeter of the region.

c) Determine the area of the region.

d) Draw a picture of the region being revolve about the x-axis.

e) What geometric figure is formed by revolving the region about the x-axis?

f) Determine the volume of the geometric solid.

g) Determine the surface area of the geometric solid.

h) If the region were rotated about the y-axis, would the volume be greater than, less than, or equal to the volume formed by revolving about the x-axis? Justify your answer.

4. A region is bounded by the graphs 5 y 3 2 yx3  (  2)2 and yx . 4 4 3 3 2 a) Draw a picture of the region. 1 x

-4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5

b) Draw a picture of the region rotated around the x-axis.

c) Draw a picture of the region rotated around the y-axis.

When drawing a sketch of a solid revolution, use the following procedure:  Draw the boundaries.  Shade the region to be revolved.  Draw the reflection (mirror image) of the region across the axis of revolution.  Connect significant points and their reflections with ellipses.