Chapter 7. Applications of the Definite integral in Geometry, Science, and Engineering

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Chapter 7. Applications of the Definite integral
in Geometry, Science, and Engineering

• By
• Jiwoo Lee
• Edited by
• Wonhee Lee

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Area Between Two Curves

• If f and g are continuous functions on the
  interval a,b and if f(x) gt g(x) for all x in
  a,b then the area of the region bounded by
  yf(x), below by g(x), on the left by the line
  xa, and on the right by the line xb is
• ?baf(x)-g(x)dx

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7.1.2 Area Formula
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Step 1

• Determine which function is on top

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Step 2Solve

• Olive green area
• ?baf(x)-g(x)dx
• Beige area ?bag(x)dx

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Tip1

• g(x) f(x)

• Sometimes it is easier to solve by integrating
  with respect to y rather than x
• ?dcf(x)-g(x)dx

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Tip2

• When finding the area enclosed by two functions,
  let the two functions equal each other and solve
  for the intersecting points to find a and b.

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Tip3

• If the two functions switch top and bottom, then
  the regions must be subdivided at those points to
  find total area

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Solve

• The area between the parabolas
• Xy2-5y and x3y-y2
• Solution
• Intersections at (0,0) and (-4,4)
• 2. Determine upper function by either plugging
  in points or graphing

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• Upper function is
• 3y-y2 , therefore
• ?403y-y2 (y2-5y)dy
• 64/3

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Solve

• the area enclosed by the two functions yx3-2x
  and y(abs(x))1/2
• Solution
• Intersection at x -1, 1.666
• Functions Switch top and bottom at x0 so the
  integral must be divided from -1 to 0 and 0 to
  1.666

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• ?0-1x3-2x-(-x)1/2dx ?1.6660(x)1/2-(x3-2x)dx
• .0832.2832.367

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7.2 Volumes by Slicing Disks and Washers

• The volume of a solid can be obtained by
  integrating the cross-sectional area from one end
  of the solid to the other.

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Volume Formula

• Let S be a solid bounded by two parallel planes
  perpendicular to the x-axis at xa and xb. If,
  for each x in a, b, the cross-sectional area of
  S perpendicular to the x-axis is A(x), then the
  volume of the solid is
• ?baA(x)dx

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• ?baA(x)dx

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Example

• The base of a solid is the region bounded by
  ye-x ,the x-axis, the y-axis, and the line x1.
  Each cross section perpendicular to the x-axis is
  a square. The volume of the Solid is...

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• V ?10 (e-x)2dx (1-1/e2)/2

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If each cross section is a circle...
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V ?ba ? (A(x))2dx

• A special case, known as method of disks,
  often used to find areas of functions rotated
  around axis or lines.

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If there are two functions rotated, then subtract
the lower region from the upper region, A method
known as method of washers

• ?ba(?(f(x))2dx- ?(g(x))2dx
• 
  f(x)
• 
  g(x)

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Rotated around a line

• A(X)

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• height A(x)-a
• Therefore, V ?ba ? (A(x)-a)2dx
• A(X)
• If a is below the x-axis, then a
  would be added to the height

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area between the two functions rotated around a
line

• G(x)
• A(x)
• ?ba(?(G(x)-a)2dx- ?(A(x)-a)2dx

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Set up but do not solve for the area using
washers method

• 1.y3x-x2 and yx rotated around the x-axis
• 2. yx2 and y4 rotated around the x-axis

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1

• V ? ?20(3x-x2)2-x2dx

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2

• V2??20(41) 2-(x21)2dx
• 2??2024-x4-2x2dx

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Volume by Cylindrical Shells

• Another method to determine the volume of a solid

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• a b

• f(x)
• a b

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• a b

• f(x)
• a b

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• f(x)
• a b

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When the section is flattened...

• width dx
• height
• f(x)
• length 2?x
• Area of Cross section 2?xf(x)dx

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Volume of f(x) rotated around the y-axis
?ba2?xf(x)dx

• f(x)
• a b

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Reminder

• Shells Method
• ?ba2?x(fx)dx
• Washers Method
• ?ba ?(f(x))2dx- ?(g(x))2dx
• When shells method is used and includes dx, then
  the function is rotated around the y axis
• When washers method is used and uses dx, then the
  function is rotated around the x axis

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Solve

• the region bounded by y3x-x2 and yx rotated
  about the y-axis

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• V ?ba2?(2x2-x3)dx 8?/3

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Length of a Plane Curve

• If f(x) is a smooth curve on the interval a,b
  then the arc length L of this curve over a,b is
  defined as
• ?bav 1 f(x)2 dx

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Length of a Plane Curve

• If no segment of the curve represented by the
  parametric equations is traced more than once as
  t increases from a to b, and if dx/dt and dy/dt
  are continuous functions for alttltb, then the arc
  length is given by
• ?bav (dx/dt) 2 (dy/dt) 2 dx

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Area of a Surface of Revolution

• If f is smooth, nonnegative function on a,b
  then the surface area S of the surface of
  revolution that is generated by revolving the
  portion of the curve y f(x) between xa and xb
  about the x-axis is defined as
• ?ba2?f(x) v1 f(x)2 dx

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Set up the integral for

• The length the curve y2x3 cut off by the line
  x4
• Solution
• 2y dy/dx 3 x2 , dy/dx (3vx)/2
• 2?40v 1 9x/4 dx

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Set up the integral for

• The surface area of y2x3 rotated around the
  x-axis from 2 to 7
• Solution y 6x2
• ?722?2x3v1 6x22dx

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Work

• If a constant force of magnitude F is applied in
  the direction of motion of an object, and if that
  object moves a distance d, then we define the
  work W performed by the force on the object to be
  
• WFd

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Work

• Suppose that an object moves in the positive
  direction along a coordinate line over the
  interval a,b while subjected to a variable
  force F(x) that is applied in the direction of
  motion. Then we define work W performed by the
  force on the object to be
• W ?baF(x)dx

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Solve

• A square box with a side length of 7 feet is
  filled with an unknown chemical. How much work is
  required to pump the chemical to a connecting
  pipe on top of the box?
• Hint The weight density of the chemical is found
  to be 90lb/ft3

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• Square box with a side length of 7
• Density of chemical found to be 90lb/ft3

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Solution

• Volume of each slice of chemical 77dy
• Increment of force 9049dy 4410dy
• Distance lifted 7-y
• Workforce distance
• 4410 ?70(7-y)dy
• 108045

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Thank you for your undivided attention