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.
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 asint_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?