MATH 2B Lecture Notes - Lecture 2: Farad, Riemann SumPremium
Course CodeMATH 2B
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MATH 2B - Lecture 2 - Areas and Distances
Area of a Rectangle
● The starting point of integral calculus is the problem of calculating area
naÏve concept of area comes from the formula for the area of rectangle
Area = Length · Width
● The area of a triangle is then immediately half that of a rectangle, and any shape
which may be subdivided into triangles may have its area computed.
Area Under the Curve
● The primary question of integral calculus is of how to extend this idea to cover
shapes which cannot be built from triangles.
● For example, below is the graph of the curve between 0 ≤ x ≤ 2;)xy = 1 + x−3
how are we to compute the shaded area?
● We approximate the area under the curve by rectangles and sum the areas of
these. If we take a larger number of rectangles, we will hopefully obtain a better
approximation to the desired area
○ Does this approximating process work for all functions?
○ Does it matter how
we choose the rectangles
● The process of this is inefficient because a
human can’t evaluate the sum of 1500 area of
Upper and Lower Bounds for Areas Example
● The area under the curve between x = 0y=x3
and x = 1 is estimated using rectangles
● Upper bound: n rectangles fitting just over curve
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