Chapter 4

Question 1

What is a Riemann sum?

Answer 1

A way to approximate the total area between a curve and the x-axis on a given range of x values

Question 2

How is a Riemann sum calculated for the function f(x) on the range [a, b]?

Answer 2

1. Select a value of N. This will be the number of subdivisions (such that a=x(0)<x(1)<…<x(i)<…<x(N-1)<x(N)=b)
2. Determine the length of each subdivision with the formula Δx(i) = x(i) - x(i-1).
3. Determine the height of each subdivision with the formula h(i) = f(x(@i)), where x(i-1)≤x(@i)≤x(i).
4. Calculate the area of each subdivision with the formula s(i) = h(i)⋅Δx(i).
5. Add up the area of all the rectangles with the formula A = {i=1, N}Σ{s(i)}.

The expanded formula is as follows:
A = {i=1, N}Σ{f(x(@i)) ⋅ [x(i) - x(i-1)]}

Question 3

What is Δx usually set to when calculating a Riemann sum?

Δx(i) = x(i) - x(i-1)

Answer 3

Δx = (b-a)/N, for evenly spaced intervals

Question 4

What is x(@i) usually set to when calculating a Riemann sum?

x(i-1)≤x(@i)≤x(i)

Answer 4

x(@i) = (x(i-1) + x(i))/2, so that it is the midpoint

Question 5

What is a definite integral?

Answer 5

A way to calculate the total area between a curve and the x-axis on a given range of x values

Question 6

How is a definite integral calculated for the function f(x) on the range [a, b]?

Answer 6

1. Determine the length of each subdivision with the formula Δx = (b-a)/N
2. Determine the position of each partition with the formula x(i) = a + i⋅Δx.
3. Determine the height of each subdivision with the formula h(i) = f(x(i)).
4. Calculate the area of each subdivision with the formula s(i) = h(i)⋅Δx.
5. Determine the area in terms of N with the formula A(N) = {i=1, N}Σ{s(i)}.
6. Take the limit of A(N) as N→+∞.

The expanded formula is as follows:
A = lim{N→+∞}{ {i=1, N}Σ{ f( x(i) )⋅((b-a)/N) }
A = lim{N→+∞}{ {i=1, N}Σ{ f(a + i⋅(b-a)/N)⋅(b-a)/N }

As N→+∞, Δx→0, thus allowing the sum of the subdivisions to become an arbitrarily close approximation of the true value of A.

Question 7

What is the Fundamental Theorem of Calculus?

Answer 7

For any function f(x) that is continuous on [a, b] and any antiderivative F(x) {F’(x)=f(x)}, then:

1. {a, b ∫ f(x) dx} = F(b) - F(a)
2. d/dx( {a, u(x) ∫ f(t) dt} ) = f(u(x)) ⋅ u’(x)

2. d/dx( {a, x)}∫{ f(t) }dt} ) = f(x)

Question 8

If f(x) ≤ g(x) on the interval a ≤ x ≤ b, what can be said about the relationship of {a, b ∫ f(x) dx} and {a, b ∫ g(x) dx}?

Answer 8

{a, b ∫ f(x) dx} ≤ {a, b ∫ g(x) dx}

Question 9

Given a function f(x), its absolute minimum N, and its absolute maximum M, what can be said about the value of {a, b ∫ f(x) dx}?

Answer 9

N(b - a) ≤ {a, b ∫ f(x) dx} ≤ M(b - a)

Question 10

How is integration by substitution performed?

Answer 10

1. Define f(x) and g(x), and set u=g(x).
2. Substitute g(x)=u and dx=du/g’(x) into the integral.
3. Evaluate the integral with respect to u.
4. Substitue u=g(x)