
Now we explore why we are interested in the area under a graph. Consider an object moving in a straight line with constant velocity v (assumed positive). The distance traveled over a time interval [t1,t2] is equal to ____, where Δt = (t_{2}t_{1}) is the _____ _____. State the appropriate distance formula
 vΔt
 time elapsed
 Distance traveled = velocity * time elapsed (vΔt)
**Equation #1

Because v is constant, the graph of velocity is a _______ line (figure 1 pg 225) and vΔt is equal to the area of the _______ region under the graph of velocity over [t_{1},t_{2}]. How do we rephrase the distance formula to better fit this concept?
 horizontal line
 rectangular region
 Distance traveled = area under the graph of velocity over [t_{1},t_{2}]
**Equation #2

There is, however, an important difference between these two equations: Eq. (1) makes sense only if velocity v is ______, whereas Eq. (2) is correct even if the velocity changes with time. Thus, the advantage of expressing distance traveled as an _____ is that it enables us to deal with much more general types of motion

To see why Eq (2) might be true in general, let's consider the case where velocity changes over time but is constant on intervals. In other words, we assume that the object's velocity _____ ______ from one interval to the next as in Figure 2 (pg 225). How would you find the distance traveled over each time interval? How would you find the total distance traveled?
 changes abruptly
 The distance traveled over each time interval is equal to the area of the rectangle above that interval
 The total distance traveled is the sum of the areas of the rectangles

What is our strategy when velocity changes continuously (Fig. 3 pg 226)? This idea leads to the concept of an ______
 To approximate the area under the graph by summing up areas of rectangles and then passing to a limit.
 integral

Distance traveled is equal to the _____ _____ the graph. It is approximated by the sum of the areas of the rectangles
 area under
 **See Fig 3 226

Our goal is to compute the area under the graph of a function f. In this section, we assume that f is _______ and _______, so that the graph of f lies above the xaxis (Fig 4 pg 226). The first step is to approximate the area using _______
 continuous and positive
 rectangles

To begin, choose a whole number N and divide [a,b] into N subintervals of _____ width as in Fig 4A (pg 226). The full interval [a,b] has width b  a, so each subinterval has width Δx = _______
 equal width
 Δx = (b  a)/N

State the right endpoints of the subintervals:
Hint, start with x_{1} =
x_{1} = a + Δx, x_{2} = a + 2Δx,..., x_{N}1 = a + (N  1)Δx, x_{N} = a + NΔx

Why is the last right endpoint x_{N} = b?
Next, as in Fig 4B (pg 226), construct, above each subinterval, a rectangle whose height is the value of f(x) at the _____ _____ of the sub interval
 Because a + NΔx = a + N ((b  a)/N) = b
 right endpoint

The sum of the areas of these rectangles provides an approximation to the area under the graph. The first rectangle has base Δx and height f(x_{1}), so its area is ______. Similarly, the second rectangle has height f(x_{2}) and area _______, etc. The sum of the areas of the rectangles is denoted ____ and is called the ____ ___________ _______ (State the initial formula).
Factoring out Δx, we obtain the formula:
 f(x_{1})Δx
 f(x_{2})Δx
 R_{N}
 Nth rightendpoint approximation
 R_{N} = f(x_{1})Δx + f(x_{2})Δx +...+f(x_{N})Δx
 R_{N} = Δx(f(x_{1}) + f(x_{2}) +...+f(x_{N}))

Explain the new formula for R_{N} after Δx is factored out
R_{N} is equal to Δx times the sum of the function values at the right endpoints of the subintervals

To summarize,
a =
b =
N =
Δx =
 a = left endpoint of interval [a,b]
 b = right endpoint of interval [a,b]
 N = number of subintervals in [a,b]
 Δx = (b  a)/N

Summation notation is a standard notation for writing _____ in compact form. How do we denote the sum of numbers a_{m},..., a_{n} (m ≤ n)?
 sums^{}
 nΣ_{j=m} a_{j} = a_{m} + a_{m + 1} +...+ a_{n}

The Greek letter Σ (capital sigma) stands for "sum," and the notation _____ tells us to start the summation at j = m and end it at j = n. For example ^{5}Σ_{j=1} j^{2} =
 ^{n}Σ_{j=m}
 1^{2} + 2^{2} + 3^{2} + 4^{2} +5^{2} = 55

In this summation, the j^{th} term is a_{j} = j^{2}. We refer to j^{2} as the _____ term. The letter j is called the ______ _____. It is also referred to as a dummy variable (why?)
 general term
 summation index
 because any other letter can be used instead. For example, the j can be a k or an m see pg 227

^{6}Σ_{k=4} (k^{2}  2k) =
^{9}Σ_{m=7} 1 =
 (4^{2}  2(4)) + (5^{2}  2(5)) + (6^{2}  2(6)) = 47
 1 + 1 + 1 = 3 (because a_{7} = a_{8} = a_{9} = 1)

The usual commutative, associative, and distributive laws of addition give us the following rules for manipulation summations:
 ^{n}Σ_{j=m} (a_{j} + b_{j}) = ^{n}Σ_{j=m} a_{j} + ^{n}Σ_{j=m} b_{j}
 ^{n}Σ_{j=m} Ca_{j} = C ^{n}Σ_{j=m} a_{j} (C any constant)
 ^{n}Σ_{j=1} C = nC (C any constant and n ≥ 1)

1) ^{5}Σ_{j=3} (j^{2} + j) =
2) ^{5}Σ_{j=3} j^{2} + ^{5}Σ_{j=3} j =
3) ^{100}Σ_{k=0} (7k^{2}  4k +9) =
What can be noted about #1 and #2
 1) (3^{2} + 3) + (4^{2} +4) + (5^{2} +5) = 62
 2) (3^{2} + 4^{2} + 5^{2}) + (3 + 4 + 5) = 62 (#1 and #2 are equivalent)
 3) ^{100}Σ_{k=0} 7k^{2} + ^{100}Σ_{k=0}(4k) + ^{100}Σ_{k=0} 9 =
 7 ^{100}Σ_{k=0} k^{2}  4 ^{100}Σ_{k=0} k + 9 ^{100}Σ_{k=0} 1

It is convenient to use summation notation when working with area approximations. For example, R_{N} is a sum with general term f(x_{j}):
The summation extends from j = 1 to j = N, so we can write R_{N} concisely as:
R_{N} = Δx[f(x_{1}) + f(x_{2}) +...+f(x_{N})]
R_{N} = Δx ^{N}Σ_{j=1} f(x_{j})

We shall make use of two other rectangular approximations. Divide [a,b] into ___ subintervals as before. In the left endpoint approximation L_{N}, the ______ of the rectangles are the values of f(x) at the left endpoints [Fig 6A pg 229]. What are these left endpoints? What are the sum areas of the leftendpoint rectangles?
 N
 heights
 x_{0} = a, x_{1} = a +Δx, x_{2} = a + 2Δx,..., x_{N}1 = a + (N  1) Δx
L _{N} = Δx(f(x _{0}) + f(x _{1}) + f(x _{2}) +...+f(x _{N}  1))

Note that both R_{N} and L_{N} have general term f(x_{j}), but the sum for L_{N} runs from _____ to ______ rather than from ______ to _____. Write in summation notation:
 j = 0 to j = N  1
 j = 1 to j = N
 L_{N} = Δx ^{N1}Σ_{j=0} f(x_{j})

In the midpoint approximation M_{N}, the _____ of the rectangles are the values of f(x) at the midpoints of the subintervals rather than at the _______. As we see in Fig 6B (pg 229) what are the midpoints?
 heights
 endpoints
 (x_{0} + x1)/2, (x_{1} + x_{2})/2,..., (x_{N1} + x_{N})/2

What is the sum of the areas of the midpoints rectangles?
How would you present this in summation notation?
 M_{N} = Δx [f((x_{0} + x_{1})/2) + f((x_{1} + x_{2})/2) +...+ f((x_{N1} + x_{N})/2)]
 M_{N} = Δx ^{N1}Σ_{j=0} f((x_{j} + x_{j+1})/2)

When f is increasing, the leftendpoint rectangles lie _____ the graph and rightendpoint rectangles lie _____ it

Figure 10 (pg 230) shows several rightendpoint approximations. Notice that the error in computing the area, corresponding to the yellow region above the graph, gets ______ as the number of rectangles increases. In fact, it appears that we can make the error as small as we please (How? and what would this entail?)
 smaller
 By making the number N of rectangles large enough. This would entail considering the limit as N ⇾ ∞, as this should give us the exact area under the curve

The error _______ as we use more rectangles
decreases

Theorem 1: If f is continuous on [a,b], then the endpoint and midpoint approximations approach one and the same limit as N →
____ (explain)
 N → ∞
 In other words, there is a value L such that:
 lim_{N → ∞} R_{N} = lim_{N → ∞} L_{N} = lim_{N → ∞} M_{N} = L
 If f(x) ≥ 0 on [a,b], we define the area under the graph over [a,b] to be L.

In Theorem 1, it is not assumed that f(x) ≥ 0. What if f(x) takes on negative values?
 The limit L no longer represents area under the graph, but we can interpret it as a "signed area,"
 **discussed in the next section

Theorem 1 can be illustrated using power sums (define)
The kth power sum is defined as the sum of the kth powers of the first N integers
**We shall use the power sum formulas for k = 1,2,3.

State 3 Power Sum examples
**bonus
 ^{N}Σ_{j=1} j = 1 + 2 +...+N = [N(N+1)/2] = N^{2}/2 + N/2
 ^{N}Σ_{j=1} j^{2} =1^{2} +2^{2} +...+ N^{2} = [N(N+1)(2N+1)/6] = N^{3}/3 + N^{2}/2 + N/6
 ^{N}Σ_{j=1} j^{3} = 1^{3} + 2^{3} +...+N^{3} = [N^{2}(N+1)^{2}/4] = N^{4}/4 + N^{3}/2 + N^{2}/4
 Bonus: ^{N}Σ_{j=1} 1 = N

Solve ^{6}Σ_{j=1} j^{2 }
1^{2} + 2^{2} + 3^{2} +4^{2} +5^{2} + 6^{2} = 6^{3}/3 + 6^{2}/2 + 6/6 = 91
**Notice the use of a Power Sum equation

