Here I got a conceptual problem rarely mentioned by my teacher.

When we try to figure the value of a definite integral, the general method is to find out the expression(value) of the indefinite integral, right?

but, when we change the dx to d(-x), a must change must be produced correspondingly, that's changing the upper limit and lower limit, right?

 

[math]

\int_a^b x \,dx

[/math]

since these are dummy variables let's keep the names separate.

 

so set y=-x, and dy=-dx

 

then the upper limit of x=b must mean y=-x=-b, and the lower limit becomes -a.

Thanks for kind reply to my question.

[math]

\int_a^b f(x) \,dx =

\int_a^b f(y) \,dy

[/math]

This is where a dummy variable could be used.

But, put y=-x and f(x)=x^2, from the formula above, We must keep the limits constant, However, this gives a different answer.Why?

The substitution y=-x gives the same answer. Just try it. It changes the integral to

 

[math]\int_{-a}^{-b}-y^2 dy[/math]

 

Why do you think you 'must keep the limits constant' (which doesn't really make sense).

couldn't y be the dummy variable?

Thanks

y is a dummy variable, so is x, that is why you can do substitutions.

If I put -x as the dummy variable, it is correct?

i.e. replace the y with -x directly without changing the limit

no, because you must substitute the dummy for the limits too

no, because you must substitute the dummy for the limits too

Why?

Isn't the formula in reply 3 true?

What are the limits they go from x=a to x=b, so if you set y=-x, then why are the limits y=a to y=b the same after this subsitution? You've just asserted that a=x=-y=-a and b=-b similarly.

 

Certainly the formula in post 3 is true, but it is substituting y=x.

 

Doing y=-x does not effect that change of integral, when f(x)=x^2, the substitution also changes the dx to a -dy, doesn't it?

 

Moral of story: don't mix and match, changing x to -y must be done at all points the x appears.

 

The fact that

 

[math]\int_a^bx^2dx=\int_a^b(-x)^2dx[/math]

 

does not happen because of a substitution, it is because x^2=(-x)^2

Doing y=-x does not effect that change of integral, when f(x)=x^2, the substitution also changes the dx to a -dy, doesn't it?

When can we use the formula in reply 3?

You don't give a formula in post three apart from one that is trivially always true

 

[math]\int_a^bf(x)dx=\int_a^bf(y)dy[/math]

 

that is gotten by subsituting y=x in to the integral.

Apart from relating y=x, what else can x be substituted?

You can set x to be equal to (pretty much) anything as long as you change the limits. A typical substitution might be something like [imath]x = \cos\theta[/imath].

okay, I guess you don't have to change the limits if you substitute the original variable back after integration

Apart from relating y=x, what else can x be substituted?

 

Absolutely any other letter, or any other reasonable function of x (ie don't pick one that is not differentiable).

 

Now, what was the point of all this?