Really need help with finding volumes under graph

[lucky_luke]

hi

Find volume of a body that we get by rotating space between graphs f(x) = -x^2 + 2 and
g(x) = |x| about x-axis.

My method is by finding volume V1 of space under graph f(x), then finding volume V2 of space under graph g(x), then substracting V = V1 - V2.

1)U1(x) = PI*{int}[0,1] (-x^2 + 2)^2)*dx
2)V1 = U(1) - U(0)

3)U2(x) = PI*{int}[0,1] (|x|)^2 * dx
4)V2 = U2(1) - U2(2)

V = V2 - V1

But the following method

2*PI*{int}[0,1] ( (-x^2 + 2)^2 - x^2 ) * dx

combines steps 1) and 3). Why is that allowed?



When computing it my way, I can visualise the process. First I draw in my mind shape of first rotated body, for which we find volume V1, then I similary find V2, then I can in my mind visually substract shape of first object from second one to get the final shape of the body.
And I do similar thing on paper, only using formulas.




But when using

2*PI*{int}[0,1] ( (-x^2 + 2)^2 - x^2 ) * dx

we skip intermediate steps and thus I have no idea how to visuallise this.

Also, I know final result is the same, but why?
My way we substracted volumes of two bodies, but what body shapes do we substract here :

U1 ... value of antiderivative 2*PI*{int}[0,1] ( (-x^2 + 2)^2 - x^2 ) * dx 
       at point 1

U2 ... value of antiderivative 2*PI*{int}[0,1] ( (-x^2 + 2)^2 - x^2 ) * dx 
       at point 0

U1 - U0 = V

?

thank you

 

[tkhunny]

1) Your first method does not give the correct answer. You've only half the volume. Since it is symmetrical about the y-axis, this is not a particularly difficult problem to fix.

2) It is only an important property of integrals. For a finite number of integrands, it does not matter if the arithmetic is inside or outside the integral.

3) It should not always matter that you can or cannot visualize this. 2D problems are the last time visualization may be simple. 3D is quite a bit more strenuous. 4D - very unlikely. It becomes more important to understand the mathematics and less important to trust your eyes.

 

[lucky_luke]

1) Your first method does not give the correct answer. You've only half the volume. Since it is symmetrical about the y-axis, this is not a particularly difficult problem to fix.

I fixed it.

1)U1(x) = PI*{int}[-1,1] (-x^2 + 2)^2)*dx
2)V1 = U(1) - U(0)

3)U2(x) = PI*{int}[-1,1] (|x|)^2 * dx
4)V2 = U2(1) - U2(2)

V = V2 - V1

2) It is only an important property of integrals. For a finite number of integrands, it does not matter if the arithmetic is inside or outside the integral.

Why doesn't it matter?

 

[tkhunny]

Like I said, it's an important property. It is easily proven by utilizing the same property for derivatives. The property for derivatives is easily proven by appealing to the difference quotient definition of the derivative.

You should be able to find these Theorems in any academic calculus book. Look for the term "Linear Combinations of Functions".