5.2 - The Definite Integral

Question 1

What is the definition of a Definite Integral?

Answer 1

if f is a function defined for a < x < b, we divide the interval [a,b] into n subintervals of equal width Δx = (b-a)/n. We let Xo (=a), X, X2…,Xn (=b) be the endpoints of these subintervals and we let X1, X2…, Xn* be any sample point s in these subintervals so, X1* lies in the ith subinterval [xi-1, xi]. Then the definite integral of f from a to b is

∫(b-a) f(x)dx = lim(n->∞) Σf(xi*)Δx

Question 2

What is the first theorem for Definite Integrals?

Answer 2

If f is continuous on [a,b] or if f has only a finite number of jump discontinuities , then f is integrable on [a,b]: that is, the definite integral ∫(b-a) f(x)dx exists

Question 3

What is the second theorem for definite Integrals?

Answer 3

if f is integrable on [a,b], then ∫(b-a) f(x)dx = lim(n->∞) Σf(xi*)Δx
where Δx = b-a/n and xi = a + iΔx

Question 4

When we use a limit to evaluate a definite integral, we need to know how to work, with sums, what are the equations?

Answer 4

1. Σ(i=1)i = n(n+1)/2
2. Σ(i=1)i^2 = n(n+1)(2n+1)/6
3. Σ(i=1)i^3 = [n(n+1)/2]^2
4. Σ(i=1)c = nc
5. Σ(i=1)cai = cΣ(i=1)ai
6. Σ(i=1)(ai + bi) = Σ(i=1)ai + Σ(i=1)bi
7. Σ(i=1)(ai - bi) = Σ(i=1)ai - Σ(i=1)bi

Question 5

What is the Midpoint Rule?

Answer 5

∫(b-a) f(x)dx = Σ(i=1)f(xi)Δx = Δx[f(xi) + … + f(xn)]
where Δx = b-a/n
and xi = 1/2(xi-1 + xi) = midpoint of [xi-1, xi)

∫(a-b) f(x)dx = -∫(b-a) f(x)dx

Question 6

What are the 5 properties of the Integral?

Answer 6

1. ∫(b-a)cdx = c(b-a), where c is any constant
2. ∫(b-a)[f(x) + g(x)]dx = ∫(b-a)f(x) + ∫(b-a)g(x)
3. ∫(b-a)cf(x) = ∫(b-a)f(x)dx, where c is any constant
4. ∫(b-a)[f(x) - g(x)]dx = ∫(b-a)f(x)dx - ∫(b-a)g(x)dx
5. ∫(c-a)f(x)dx + ∫(b-c)f(x)dx = ∫(b-a)f(x)dx

Question 7

What are the 3 Comparison Properties of the Integral?

Answer 7

1. If f(x)>0 for a< x <b>0</b>
2. If f(x) > g(x) for a < x <b> ∫(b-a)g(x)dx</b>
3. If m < f(x) < M for a < x < b then…
   m(b-a) < ∫(b-a)f(x)dx < M(b-a)</b></b>