Hi, I was wondering how transformations, such as dilations, affect definite integrals. Specifically, why is the first statement in the example below true? Why is the first integral 1/3 of the second? Thanks!

The first statement is this

int_1^3f(3x+1)dx=(1/3)*int_4^10f(x)dx

I am given this information

int_4^10f(x)dx=3

Therefore

int_1^3f(3x+1)dx=(1/3)*3=1

There is a dilation of 1/3 from the y-axis and a horizontal translation of 1/3 to the left, but I am not sure why the first integral is 1/3 of the second.

1 Answer
Sep 25, 2017

We aim to show that:

I = int_1^3f(3x+1)dx = (1/3)*3=1

Given:

int_4^10 f(x) \ dx = 3

We can perform a substitution on the first integral:

Let u=3x+1 => (du)/dx = 3

And when we perform a transformation substitution, we must change the limits of integration accordingly,

When x = { (1), (3) :} => u = { (4), (10) :}

Now, we can prepare and perform the substitution, and change the integration limits, which gives:

I = int_1^3 \ f(3x+1) \ dx
\ \ = int_1^3 \ f(3x+1) \ (1/3)(3)dx
\ \ = int_4^10 \ f(u) \ (1/3) \ du
\ \ = 1/3 \ int_4^10 \ f(u) \ du \ \ \ \ ..... (A)
\ \ = 1/3 (3) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ..... (B)
\ \ = 1 QED

Here we have used two fundamental properties of integrals, namely:

Scaling by a Constant

int \ c \ f(t) \ dt = c \ int \ f(t) \ dt where #\# is constant

This was used at [A] above to factor out the 1/3 multiplier.

Changing the variable alone of a definite Integration

int_a^b \ f(x) \ dx = int_a^b \ f(z) \ dz = int_a^b \ f(t) \ dt

This was used at [B] above to evaluate:

int_4^10 \ f(u) du \ \ \ , which is the same as int_4^10 \ f(x) dx

Not that here no transformation has occurred. If we think about a definite integral in term of the area under the curve, we are simply saying that if we change the label along on the graph of the function fromx to t, say the area does not change.

Further Explanation:

For further insight as to why the variables must be changed during a transformation, please review this solution:

Evaluating definite Integral and changing the values of A and B?