# 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!

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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.

The first statement is this

I am given this information

Therefore

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

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

**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 from

**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?