Areas of Domains and Definite Integrals

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Areas of Domains and Definite Integrals

• Area under a ParabolaDefinition of the Area of
  Certain DomainsArea under a Parabola
  RevisitedIntegrals and Antiderivatives

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Estimate Areas
Consider the problem of determining the area of
the domain bounded by the graph of the function
x2, the x-axis, and the lines x0 and x1.
We determine the area by approximating the domain
with thin rectangles for which the area can be
directly computed. Letting these rectangles get
thinner, the approximation gets better and, at
the limit, we get the area of the domain in
question.
As the number n of the approximating rectangles
grows, the approximation gets better.
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Estimate Areas (2)
Let A denote the actual area of the domain in
question. Clearly snltA for all n.
Lower est. sn
Upper est. Sn
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Estimate Areas (3)
This can be computed directly using a previously
derived formula for the sum of squares. Solution
follows.
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Estimate Areas (4)
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Conclude
The blue area under the curve yx2 over the
interval 0,1 equals 1/3.
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Definition of the Area of a Domain under the
Graph of a Function
The previous considerations were based on some
intuitive idea about areas of domain. We make
that precise in the statement of the following
result.
Theorem
Definition
The common value of these limits is the area of
the domain under the graph of the function f and
over the interval a,b.
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The Integral (1)
Theorem
Definition
The common value of these limits is the integral
of the function f over the interval a,b.
Notation
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The Integral (2)
Remark
This means that the points xk may be freely
selected from the intervals a (k-1)/n, a
k/n. The limit of the sum does not depend on
the choice of the points xk.
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The Integral (3)
Remark
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Examples (1)
Example 1
Solution
By the Definition
Now use the formula for the sum of squares
Conclude
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Examples (2)
Example 2
Solution
The red area under the graph of x2 over a,b
equals the area over 0,b minus the area over
0,a.
Conclude
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Integrals and Antiderivatives
Rewriting the previous result in the following
form
we observe that the integral defines the function

Direct differentiation yields F(x) x2, i.e.,
the function F is an antiderivative of the
function f(x) x2.
A proof of this result will be presented later.
Theorem