Topic 9: Integration

Question 1

What is the mean value theorem for definite integrals (1)

Answer 1

Let f: ℝ -> ℝ be continuous on [a, b]
-Then, there exists c ∈ (a, b) such that ∫^b_a f(x) dx = (b-a)f(c)

Question 1

How does the fundamental theorem of calculus say integration and differentiation are inverse operations (2,2,1)

Answer 1

Let f: ℝ -> ℝ
-if f is Riemann-integrable over (a, b) and F(x) = ∫^x_a f(t) dt, then F is a continuous function of x on [a, b]
-Furthermore, if f is continuous on [a, b], then F is differentiable and F’ = f

In this case:
-∫^b_a f(x) dx = F(b) - F(a)
-d/dx ∫^x_a f(t) dt = f(x)

-Therefore, integration and differentiation are inverse operations

Question 2

How can we attach a rigorous meaning to ∫^∞_a f(x) dx (2)

Answer 2

-Define ∫^∞_a f(x) dx = lim_b->∞ ∫^b_a f(x) dx
-Compare this with the convergence of the series, as ∫ is the continuous analogue of ∑

Question 3

How can we calculate ∫^∞₀ e^-x dx (3,1)

Answer 3

-∫^∞₀ e^-x dx = lim_b->∞ ∫^b₀ e^-x dx
=lim_b->∞[-e^-x]^b₀
=lim_b->∞[-e^-b + 1] = 1

-Therefore, ∫^∞₀ e^-x dx = 1

Question 4

How can we solve ∫^∞_-∞ f(x) dx (2)

Answer 4

-∫^∞_-∞ f(x) dx = ∫^a_-∞ f(x) dx + ∫^∞_a f(x) dx
-Not ∫^∞_-∞ f(x) dx = ∫^b_-b f(x) dx

Question 5

What are 3 types of unbounded intervals (1, 3)

Answer 5

-Say we want to evaluate ∫^b_a f(x) dx, where f(x) is unbounded somewhere in [a, b]

3 Cases:
-f(x) becomes infinite at x = a
-f(x) becomes infinite at x = b
-f(x) becomes infinite at x = c ∈ (a, b)

Question 6

How do you evaluate ∫^b_a f(x) dx if f(x) becomes infinite at x = a (2)

Answer 6

-∫^b_a f(x) dx = lim_h->0⁺ ∫^b_a+h f(x) dx
=lim_c->a⁺ ∫^b_c f(x) dx

Question 7

How do you evaluate ∫^b_a f(x) dx if f(x) becomes infinite at x = b (2)

Answer 7

-∫^b_a f(x) dx = lim_h->0⁺ ∫^b-h_a f(x) dx
=lim_c->b⁺ ∫^c_a f(x) dx

Question 8

How do you evaluate ∫^b_a f(x) dx if f(x) becomes infinite at x = c ∈ (a, b) (2)

Answer 8

-∫^b_a f(x) dx = ∫^c_a f(x) dx + ∫^b_c f(x) dx
-Then solve this like the first 2 cases

Question 9

What is the formula for integration by parts (1)

Answer 9

-∫u(dv) dx = uv - ∫(v)(du) dx