Integration

Question 1

Definite Integral

Answer 1

the integral of a function over a set range

Question 2

Integral of x^m between 0 and b

Answer 2

b^(m+1) / (m+1)

Question 3

Upper Riemann Sum

Answer 3

• used to find an estimate for the integral of a function
• the function is split in to n vertical strips
• for the upper sum rectangles are taken above the line of the function
• gives an over estimate

Question 4

Lower Riemann Sums

Answer 4

• used to find an estimate for the integral of a function
• the function is split into n vertical strips
• for the lower sum take rectangles that are below the line of the function
• gives an under estimate

Question 5

First Fundamental Theorem of Calculus

Answer 5

Let f be integratable on [a,b] and define F(x) as the integral of f(t) dt. If f is continuous at c, then F is differentiable at c and
F’(c) = f(c)

Question 6

Second Fundamental Theorem of Calculus

Answer 6

If f I continuous on [a,b] and G is any function such that G’ = f, then:
the integral of f between a and b = G(b) - G(a)

Question 7

Infinite and Unbounded Integrals

Answer 7

• if substituting one of the limit values into the function gives infinity
• replace that limit with ε and find the limit of the integral as ε->limit

Question 8

Integration by Parts

Answer 8

∫uv’ = uv - ∫v*u’

To use this, vu’ must be easier to integrate than uv’

Question 9

Integration by Substitution

Answer 9

1) to find ∫f(x) dx, substitute
u = u(x)

2) to find ∫f(x) dx, substitute
x = x(u)