INTEGRALS

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INTEGRALS
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INTEGRALS
5.3The Fundamental Theorem of Calculus
In this section, we will learn about The
Fundamental Theorem of Calculus and its
significance.
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FUNDAMENTAL THEOREM OF CALCULUS

• The Fundamental Theorem of Calculus (FTC) is
  appropriately named.
• It establishes a connection between the two
  branches of calculusdifferential calculus and
  integral calculus.

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FTC

• Differential calculus arose from the tangent
  problem.
• Integral calculus arose from a seemingly
  unrelated problemthe area problem.

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FTC

• The FTC gives the precise inverse relationship
  between the derivative and the integral.

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FTC
Equation 1

• The first part of the FTC deals with functions
  defined by an equation of the form
• where f is a continuous function on a, b and x
  varies between a and b.

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FTC

• Observe that g depends only on x, which appears
  as the variable upper limit in the integral.
• If x is a fixed number, then the integral
  is a definite number.
• If we then let x vary, the number
  also varies and defines a function of x denoted
  by g(x).

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FTC

• If f happens to be a positive function, then g(x)
  can be interpreted as the area under the graph of
  f from a to x, where x can vary from a to b.
• Think of g as the area so far function, as
  seen here.

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FTC
Example 1

• If f is the function whose graph is shown and
  , find the values of
  g(0), g(1), g(2), g(3), g(4), and g(5).
• Then, sketch a rough graph of g.

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FTC
Example 1

• First, we notice that

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FTC
Example 1

• From the figure, we see that g(1) is the area of
  a triangle

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FTC
Example 1

• To find g(2), we add to g(1) the area of a
  rectangle

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FTC
Example 1

• We estimate that the area under f from 2 to 3 is
  about 1.3.
• So,

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FTC
Example 1

• For t gt 3, f(t) is negative.
• So, we start subtracting areas, as follows.

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FTC
Example 1

• Thus,

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FTC
Example 1

• We use these values to sketch the graph of g.
• Notice that, because f(t) is positive for t lt 3,
  we keep adding area for t lt 3.
• So, g is increasing up to x 3, where it
  attains a maximum value.
• For x gt 3, g decreases because f(t) is negative.

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FTC

• If we take f(t) t and a 0, then, using
  Exercise 27 in Section 5.2, we have

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FTC

• If we sketch the derivative of the function g, as
  in the first figure, by estimating slopes of
  tangents, we get a graph like that of f in the
  second figure.
• So, we suspect that g f in Example 1 too.

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FTC

• To see why this might be generally true, we
  consider a continuous function f with f(x) 0.
• Then, can be
  interpreted as the area under the graph of f
  from a to x.

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FTC1

• If f is continuous on a, b, then the function g
  defined by
• is continuous on a, b and differentiable on
  (a, b), and g(x) f(x).

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FTC1

• In words, the FTC1 says that the derivative of a
  definite integral with respect to its upper limit
  is the integrand evaluated at the upper limit.

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FTC1
Equation 5

• Using Leibniz notation for derivatives, we can
  write the FTC1 as
• when f is continuous.
• Roughly speaking, Equation 5 says that, if we
  first integrate f and then differentiate the
  result, we get back to the original function f.

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FTC1
Example 2

• Find the derivative of the function
• As is continuous, the FTC1 gives

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FTC1
Example 3

• A formula of the form may seem like a
  strange way of defining a function.
• However, books on physics, chemistry, and
  statistics are full of such functions.

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FTC1
Example 4

• Find
• Here, we have to be careful to use the Chain Rule
  in conjunction with the FTC1.

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FTC1
Example 4

• Let u x4.
• Then,

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FTC1

• In Section 5.2, we computed integrals from the
  definition as a limit of Riemann sums and saw
  that this procedure is sometimes long and
  difficult.
• The second part of the FTC (FTC2), which follows
  easily from the first part, provides us with a
  much simpler method for the evaluation of
  integrals.

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FTC2

• If f is continuous on a, b, then
• where F is any antiderivative of f, that is, a
  function such that F f.

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FTC2

• The FTC2 states that, if we know an
  antiderivative F of f, then we can evaluate
  simply by
  subtracting the values of F at the endpoints of
  the interval a, b.

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FTC2

• Its very surprising that , which
  was defined by a complicated procedure involving
  all the values of f(x) for a x b, can be
  found by knowing the values of F(x) at only two
  points, a and b.

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FTC2

• If v(t) is the velocity of an object and s(t) is
  its position at time t, then v(t) s(t).
• So, s is an antiderivative of v.

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FTC2

• In Section 5.1, we considered an object that
  always moves in the positive direction.
• Then, we guessed that the area under the velocity
  curve equals the distance traveled.
• In symbols,
• That is exactly what the FTC2 says in this
  context.

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FTC2
Example 5

• Evaluate the integral
• The function f(x) x3 is continuous on -2, 1
  and we know from Section 4.9 that an
  antiderivative is F(x) ¼x4.
• So, the FTC2 gives

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FTC2
Example 5

• Notice that the FTC2 says that we can use any
  antiderivative F of f.
• So, we may as well use the simplest one, namely
  F(x) ¼x4, instead of ¼x4 7 or ¼x4 C.

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FTC2

• We often use the notation
• So, the equation of the FTC2 can be written as
• Other common notations are and
  .

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FTC2
Example 6

• Find the area under the parabola y x2 from 0 to
  1.
• An antiderivative of f(x) x2 is F(x) (1/3)x3.
  
• The required area is found using the FTC2

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FTC2
Example 7

• Find the area under the cosine curve from 0 to b,
  where 0 b p/2.
• Since an antiderivative of f(x) cos x is F(x)
  sin x, we have

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FTC2
Example 7

• In particular, taking b p/2, we have proved
  that the area under the cosine curve from 0 to
  p/2 is sin(p/2) 1.

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FTC2

• If we didnt have the benefit of the FTC, we
  would have to compute a difficult limit of sums
  using either
• Obscure trigonometric identities
• A computer algebra system (CAS), as in Section 5.1

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FTC2
Example 8

• What is wrong with this calculation?

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FTC2
Example 9

• To start, we notice that the calculation must be
  wrong because the answer is negative but f(x)
  1/x2 0 and Property 6 of integrals says that
  when f 0.

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FTC2
Example 9

• The FTC applies to continuous functions.
• It cant be applied here because f(x) 1/x2 is
  not continuous on -1, 3.
• In fact, f has an infinite discontinuity at x
  0.
• So, does not exist.

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INVERSE PROCESSES

• We end this section by bringing together the two
  parts of the FTC.

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FTC

• Suppose f is continuous on a, b.
• 1.If , then g(x) f(x).
• 2. , where F is any antiderivative
  of f, that is, F f.

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INVERSE PROCESSES

• We noted that the FTC1 can be rewritten as
• This says that, if f is integrated and then the
  result is differentiated, we arrive back at the
  original function f.

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INVERSE PROCESSES

• As F(x) f(x), the FTC2 can be rewritten as
• This version says that, if we take a function F,
  first differentiate it, and then integrate the
  result, we arrive back at the original function
  F.
• However, its in the form F(b) - F(a).

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INVERSE PROCESSES

• Taken together, the two parts of the FTC say that
  differentiation and integration are inverse
  processes.
• Each undoes what the other does.

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SUMMARY

• The FTC is unquestionably the most important
  theorem in calculus.
• Indeed, it ranks as one of the great
  accomplishments of the human mind.