Mean Value Theorem for Integrals and Monte Carlo Integration

function

x

3

-5x

2

-8x+3

a

-1.69

b

2.55

n

563

random seed

263

-3

-2

-1

1

2

3

-50

-40

-30

-20

-10

∫

b

a

x

3

-5x

2

-8x+3 dx = -29.0146

Monte Carlo estimate: -30.9877

The mean value theorem for integrals states that if

f(x)

is continuous over

[a,b]

, then there exists a real number

c

with

a<c<b

such that

1

b-a

∫

b

a

f(x)dx=f(c)

. Writing this as

∫

b

a

f(x)dx=(b-a)f(c)

shows that the area under the curve is the base

(b-a)

times the "average height"

f(c)

. To estimate this integral by the Monte Carlo method, use the following steps:

(1) Pick

n

uniformly distributed numbers

x

1

,x

2

,…,x

n

in the interval [

a

,

b

].

(2) Evaluate the function

f

at each point and calculate the average function value:

-

f

=

1

n

∑

i=1

f(x

i

)

.

(3) Compute the approximate value of the integral:

(b-a)×

-

f

.