5.4 Theorems About Definite Integrals

Question 1

What is the Fundamental Theorem of Calculus (Part 1)?

Answer 1

If f is continuous on [𝑎,𝑏] and F(x) is an antiderivative of f(x), then:
∫ (b on top a on bottom) f(x)dx=F(b)−F(a)

Question 1

What is the Fundamental Theorem of Calculus (Part 2)?

Answer 1

If f is continuous on [𝑎,𝑏] and F(x) is defined by ∫ (b on top a on bottom) f(t)dt, then F(x) is an antiderivative of f(x), meaning f(x)=F’(x).

Question 2

What is the Additivity of Integrals Theorem?

Answer 2

If f is continuous on [𝑎,𝑏], and c is a point in [a,b] then:
∫ (b on top a on bottom)f(t)dx=∫ (c on top a on bottom)f(t)dx+ ∫(b on top c on bottom)f(t)dx

Question 2

What is the Zero Width Interval Theorem?

Answer 2

If a=b, then the definite integral over that interval is zero:
∫ (a on top a on bottom)f(x)dx=0

Question 3

What is the Integral of a Constant Multiple Theorem?

Answer 3

If k is a constant and f(x) is integrable [a,b], then:

∫ (b on top a on bottom) kf(x)dx=k∫ (b on top a on bottom)f(x)dx

Question 4

What is the Sum of Integrals Theorem?

Answer 4

If f(x) and g(x) are integrable on
[a,b], then:

∫ (b on top a on bottom) [f(x)+g(x)]dx= ∫ (b on top a on bottom) f(x)dx+ ∫ (b on top a on bottom) g(x)dx

Question 5

What is the Comparison Theorem?

Answer 5

If f(x)≤g(x) for all 𝑥 in [a,b], and both functions are continuous, then:

∫ (b on top a on bottom) f(x)dx ≤ ∫ (b on top a on bottom) g(x)dx