Chapter 5

Question 1

Let f(x) and g(x) be continuous and f(x) ≥ g(x) for all a ≤ x ≤ b. What is the area of the region bounded by the curves y=f(x), y=g(x), x=a, and x=b?

Answer 1

{a, b ∫ {f(x) - g(x)} dx}

Question 2

How do you find the area of the region bounded by two curves y=f(x) and y=g(x)? (Assume f(x) ≥ g(x) on the relevant values of x.)

Answer 2

1. Find two points where f(x) = g(x), and define a as the smaller of the two and b as the larger of the two.
2. Evaluate {a, b ∫ {f(x) - g(x)} dx}.

Question 3

How is u-substitution performed?

Answer 3

1. Define u(x) in terms of x
2. Find du(x) = d/dx (u(x)) ⋅ dx
3. Substitute u(x) and du(x) into the integral
4. Replace a and b (the limits of integration) with u(a) and u(b)

Question 4

What are the two main ways a revolved solid can be split up to find its volume?

Answer 4

• Washers/disks
• Cylindrical shells

Question 5

How do you calculate the average of a function f(x) over an interval [a, b]?

Answer 5

{a, b ∫ {f(x)} dx} / (b - a)

Question 6

What is the Mean Value Theorem for integrals?

Answer 6

If f(x) is continuous on a≤x≤b, then there exists a number c such that a≤c≤b and f(c) = {a, b ∫ {f(x)} dx} / (b - a)