print barcode image c# The Integral in Software

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CHAPTER 4 The Integral
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For our rst example, we calculate the area under a parabola EXAMPLE 44
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Calculate the area under the curve y = x 2 , above the x-axis, and between x = 0 and x = 2
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SOLUTION Refer to Fig 48 as we reason along Let f (x) = x 2
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Fig 48
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Consider the partition P of the interval [0, 2] consisting of k + 1 points x0 , x1 , , xk The corresponding Riemann sum is
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R(f, P ) =
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j =1
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f (xj )
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Of course x= and 2 xj = j k In addition, f (xj ) = j 2 k
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2 2 0 = k k
4j 2 k2
As a result, the Riemann sum for the partition P is
The Integral
R(f, P ) =
j =1 k
4j 2 2 k2 k 8 8j 2 = 3 3 k k
j =1
j 2
j =1
Now formula II above enables us to calculate the last sum explicitly The result is that
R(f, P ) =
8 2k 3 + 3k 2 + k 6 k3 4 8 4 = + + 2 3 k 3k 4 8 8 4 + + 2 = 3 k 3k 3
x 2 dx = lim R(f, P ) = lim
We conclude that the desired area is 8/3
The most important idea in all of calculus is that it is possible to calculate an integral without calculating Riemann sums and passing to the limit This is the Fundamental Theorem of Calculus, due to Leibniz and Newton We now state the theorem, illustrate it with examples, and then brie y discuss why it is true Theorem 41 (Fundamental Theorem of Calculus) Let f be a continuous function on the interval [a, b] If F is any antiderivative of f then
You Try It: Use the method presented in the last example to calculate the area under the graph of y = 2x and above the x-axis, between x = 1 and x = 2 You should obtain the answer 3, which of course can also be determined by elementary considerations without taking limits
EXAMPLE 45
Calculate
AM FL Y
In sum,
f (x) dx = F (b) F (a)
x 2 dx
CHAPTER 4 The Integral
SOLUTION We use the Fundamental Theorem In this example, f (x) = x 2 We need to nd an antiderivative F From our experience in Section 41, we can determine that F (x) = x 3 /3 will do Then, by the Fundamental Theorem of Calculus,
x 2 dx = F (2) F (0) =
23 03 8 = 3 3 3
Notice that this is the same answer that we obtained using Riemann sums in Example 44 EXAMPLE 46
Calculate
sin x dx
SOLUTION In this example, f (x) = sin x An antiderivative for f is F (x) = cos x Then
sin x dx = F ( ) F (0) = ( cos ) ( cos 0) = 1 + 1 = 2
EXAMPLE 47
Calculate
ex cos 2x + x 3 4x dx
SOLUTION In this example, f (x) = ex cos 2x + x 3 4x An antiderivative for f is F (x) = ex (1/2) sin 2x + x 4 /4 2x 2 Therefore
ex cos 2x + x 3 4x dx = F (2) F (1) = e2 24 1 sin(2 2) + 2 22 2 4
14 1 sin(2 1) + 2 12 2 4 9 1 = (e2 e) [sin 4 sin 2] 4 2 e1
You Try It: Calculate the integral
1 3
The Integral
x 3 cos x + x dx
Math Note: Observe in this last example, in fact in all of our examples, you can use any antiderivative of the integrand when you apply the Fundamental Theorem of Calculus In the last example, we could have taken F (x) = ex (1/2) sin 2x + x 4 /4 2x 2 + 5 and the same answer would have resulted We invite you to provide the details of this assertion Justi cation for the Fundamental Theorem Let f be a continuous function on the interval [a, b] De ne the area function F by F (x) = area under f , above the x-axis, and between 0 and x
Fig 49
Let us use a pictorial method to calculate the derivative of F Refer to Fig 49 as you read on Now F (x + h) F (x) [area between x and x + h, below f ] = h h f (x) h h = f (x) As h 0, the approximation in the last display becomes nearer and nearer to equality So we nd that
F (x + h) F (x) = f (x) h
But this just says that F (x) = f (x) What is the practical signi cance of this calculation Suppose that we wish to calculate the area under the curve f , above the x-axis, and between x = a and x = b Obviously this area is F (b) F (a) See Fig 410 But we also know that
CHAPTER 4 The Integral
that area is
b a f (x) dx
We conclude therefore that
f (x) dx = F (b) F (a)
y = f (x)
F (b) _ F (a)
Fig 410
Finally, if G is any other antiderivative for f then G(x) = F (x) + C Hence G(b) G(a) = [F (b) + C] [F (a) + C] = F (b) F (a) =
f (x) dx
That is the content of the Fundamental Theorem of Calculus You Try It: Calculate the area below the curve y = x 2 + 2x + 4 and above the x-axis
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