6.1: Areas Between Curves

Question 1

What is the Definition of Areas Between the Curves?

Answer 1

The area A of the region bounded by the curves y = f(x) lines x=a, x=b where f and g are continuos & f(x) > g(x) for all x in [a,b] is

A = ∫(b-a) [f(x) - g(x)]dx

Question 2

What is the equation for Average value of a Function?

Answer 2

fave = 1/(b-a) ∫(b-a) f(x)dx

Question 3

What is the Mean Value Theorem for Integrals?

Answer 3

If f is continuous on [a,b], then there exists a number c in [a,b] such that
f(c) = fave = 1/(b-a) ∫(b-a) f(x)dx that is,
∫(b-a) f(x)dx = f(c)(b-a)

Question 4

Trigonometric Integrals: Strategy for Evaluating ∫sin^m(x)cos^n(x)dx ?

Answer 4

1. If the power of cosine is odd (n = 2k+1), same one cosine factor & use cos^2(x) = 1 - sin^2(x) to express the remaining factors…..Then substitute u = sin(x)
2. If the power of sine is odd (m = 2k + 1), save one factor & use sin^2(x) = 1 - cos^2(x) to express the remaining factors in terms of cosine ….Substitute u =cos(x)
3. If the powers of both sine and cosine are even, use the half-angle identities
   sin^2(x) = 1/2 (1 - cos(2x))
   cos^2(x) = 1/2 (1 + cos(2x))

Question 5

For Trigonometric Identities, it is sometimes helpful to use which identity?

Answer 5

sin(x)cos(x) = 1/2 sin(2x)

Question 6

Trigonometric Integrals: Strategy for Evaluating ∫tan^m(x)sec^n(x)dx?

Answer 6

1. If the power of secant is even (n = 2k, k>2), save a factor of sec^2(x) and use sec^2(x) = 1 + tan^2(x) to express the remaining factors in terms of tan(x)
   Then Substitute U = tan(x)
2. If the power of tangent is odd (m = 2k +1), save a factor of the sec(x)tan(x) and use TAN^2(x) = SEC^2(x) - 1 to express the remaining factors in terms of sec(x)
   Then substitute U = sec(x)

Question 7

What is the antiderivative of ∫tan(x)dx ?

Answer 7

ln|sec(x)| + C

Question 8

What is the antiderivative of ∫sec(x)dx?

Answer 8

ln|sec(x) + Tan(x)| + C

Question 9

For the Trigonometric Substitution Expression

√a^2 - x^2 what is the substitution?

Answer 9

x = asinθ, -π/2

Question 10

For the Trigonometric Substitution Expression

√a^2 + x^2 what is the substitution?

Answer 10

x = atanθ, -π/2

Question 11

For the Trigonometric Substitution Expression

√x^2 - a^2 what is the substitution?

Answer 11

x = asecθ, 0