Area

Sigma Notation This section begins by introducing a concise notation for sums. This notation is called sigma notation because it uses the uppercase Greek letter sigma, written as

Example 1 – Examples of Sigma Notation From parts (a) and (b), notice that the same sum can be represented in different ways using sigma notation.

Sigma Notation The following properties of summation can be derived using the Associative and Commutative Properties of Addition and the Distributive Property of Addition over Multiplication. (In the first property, k is a constant.)

Sigma Notation The following theorem lists some useful formulas for sums of powers.

Example 2 – Evaluating a Sum

How could you find the area under the curve y=x2 from [ 0, 1]?? - Can think of total area under the curve as the sum of the areas of rectangles making up the curve - As the rectangles get smaller the approximation of the area will be closer to the true area

Rectangles can be drawn from the right side or left side of the curve The true area will be somewhere between the right and left end areas

Example Find the area under the curve y = x2 from [ 0,1 ] using right and left end rectangles of width ¼

Improving Accuracy Better estimates of the area can be obtained by using more rectangles As the number of rectangles used approaches the estimate approaches the true value of the area Since both the right and left rectangles approach the same value, it doesn’t matter which is used

General Form For a function f(x) on interval of [a,b] - the width of one rectangle can be given by Δx and the height by f(xi)

Example Find the following limit

Example Find the area of the region bounded by the curve f(x)=x2 between x=0 and x=2

Example Find the area of the region bounded by the curve f(x) = x2 + 1 between x =0 and x = 2

Example Find the area of the region bounded by the curve f(x) = 3x-4 between x=2 and x=5

Example Find the area bounded by f(y) = y2 for 0 ≤ y ≤ 1