7.3 Volumes - Oxford School District

7.3 Volumes - Oxford School District

7.3 VOLUMES WITH KNOWN CROSS SECTIONS VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. SOLIDS WITH KNOWN CROSS SECTIONS If A(x) is the area of a cross section of a solid and A(x) is continuous on [a, b], then the volume of the solid from x = a to x = b is b V A( x)dx a VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. 3

-3 y 3 -3 dx Area of cross section (square)? 2 A s 2 A (2 y ) 2 A 4 y 2 x y 9 y 9 x y-coordinate So, s = 2y A 4 9 x x

2 2 A 4(9 x ) 2 2 2 VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. 3 -3 Area of cross section 2 y (square)? A 4(9 x )

Volume of solid: x2 V A( x)dx x1 3 -3 dx x 3 V 4(9 x 2 )dx 3 VOLUMES WITH KNOWN CROSS SECTIONS A solid has as its base the circle x2 + y2 = 9, and all cross sections parallel to the y-axis are squares. Find the volume of the solid. Volume of solid:3 3 -3 V 4(9 x 2 )dx

y 3 3 V 4 (9 x 2 )dx 3 1 3 V 4 9 x x 3 3 3 -3 dx 3 x V 418 18 V 4 36 V 144 KNOWN CROSS SECTIONS Ex: The base of a solid is the region enclosed by x2 y2 the ellipse

1 4 25 The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid. 5 -2 a a 2 -5 5 1.) Find the area of the cross section A(x). -2 a a 2 2 a a (2 y ) 2 2 2a 4 y a 2 y

y 2 2 -5 1 2 A( x) a 2 1 A( x) 2 y 2 A( x) y 2 2.) Set up & evaluate the integral. 2 25 x 2 A( x) 25 4 2 2 200

25 x 3 units 25 dx 3 4 2 EXAMPLE The base of a solid is the region enclosed by the triangle whose vertices are (0, 0), (4, 0), and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Find the volume of the solid.y Area of cross section (semicircle)? 1 2 A r 2 2 4 1 1 1 A x 2

2 2 2 x r is half of the yvalue on the line y mx b 2 1 1 A x 1 2 4 y 2 1 x2 2 EXAMPLE The base of a solid is the region enclosed by the triangle whose vertices are (0, 0), (4, 0), and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Find the volume of the solid.y Area of cross section (semicircle)?

1 1 A x 1 2 4 2 Volume 4 2 2 1 1 V x 1 dx 2 0 4 4 x (fInt) V = 2.094

Recently Viewed Presentations

  • adta.org

    adta.org

    Moving with Mary influenced me deeply. We went to a deep place together. It wasn't moving to a beautiful position or anything like that, it was finding something deep inside which eventually we called an authentic place. She was working...
  • The characteristics, objectives and benefits of Leisure ...

    The characteristics, objectives and benefits of Leisure ...

    Leisure time Play, physical recreation and sport. Characteristics 'Spare time' during which you choose what to do- Personal Choice. Available once all other commitments are completed. E.g. Going to school ,part-time job, homework etc. Various ways to spend it. Some...
  • A powerful tool that automates and grows your

    A powerful tool that automates and grows your

    CRM + Automated Marketing + Tools Integration. The all-new Royal LePage Smart Studio integrates a CRM and other critical business and marketing functions to automate the work you do.
  • Artist: 10 - Weebly

    Artist: 10 - Weebly

    Randy Howell. October 12, 2015. J. Flowers. How to launch the communication plan. ... (Tong, 2015, p.1) The Effectiveness of the Communication Plan. Ensure employees understand how the changes will influence their future. Help to prepare for future changes.
  • Argument Analysis: Claim - Reason - Foundation

    Argument Analysis: Claim - Reason - Foundation

    Argument Analysis. The very first thing to do when beginning an analysis is to put away your biases (at least as much as you can). It doesn't really matter what you think about the issue or the argument; in fact,...
  • Science Today

    Science Today

    5. Their beaks were different. 6. obtaining food. 7. D. 8. C. 9. The finches were descended from South American finches blown to the islands by a storm. Over many generations, the finches adapted to life on the island. 10....
  • Introduction to Torts

    Introduction to Torts

    There is causation in fact for some of these injuries, but, for example, there is no causation in fact for Eric's wife's lightning strike-related injuries. Proximate cause is also much less likely to be found for the lightning strike injuries,...
  • u

    u

    UCAS Progress. Search and Apply function for Year 10 and 11 students. Current national roll-out project underway. With all young people under 18 in England now required to stay in education or work-based learning, UCAS Progress offers an important source...