If f(x) and g(x) are continuous on [a, b], which one of the following statements is true?
~the integral from a to b of the difference of f of x and g of x, dx equals the integral from a to b of f of x, dx minus the integral from a to b of g of x dx
~the integral from a to a of f of x, dx equals 2 times the integral from 0 to a of f of x, dx
~the integral from a to b of the product of f of x and g of x, dx equals the integral from a to b of f of x, dx times the integral from a to b of g of x dx
(MY ANSWER)
~the integral from a to b of 3 plus f of x, dx equals 3 plus the integral from a to b of f of x, dx
Well, well, well, look who we have here - a question about integrals! Let's see if we can find the right answer to this one.
The statement you picked, my friend, is not quite accurate. The integral from a to b of the difference of f(x) and g(x), dx, does not equal the integral from a to b of f(x), dx minus the integral from a to b of g(x), dx. That's not how integration works!
As for the second statement, it's not true either. The integral from a to a of f(x), dx is always zero, no matter what f(x) is. So it definitely ain't equal to 2 times the integral from 0 to a of f(x), dx. Nice try though!
Now, let's move on to the fourth statement. Ah, I see your wit in adding a little extra to the integral! But alas, it's not the correct answer. The integral from a to b of 3 plus f(x), dx is indeed 3 times the integral from a to b of f(x), dx, but it's not "3 plus" the integral from a to b of f(x), dx.
So, my friend, that leaves us with only one statement left - the third one. It's the winner! The integral from a to b of the product of f(x) and g(x), dx does indeed equal the integral from a to b of f(x), dx times the integral from a to b of g(x), dx. You got it!
Now, go forth and spread the wisdom of integrals and math humor!
The correct statement is:
~the integral from a to b of the product of f(x) and g(x), dx equals the integral from a to b of f(x), dx times the integral from a to b of g(x), dx
To determine which statement is true, let's examine each option:
1. The statement "the integral from a to b of the difference of f of x and g of x, dx equals the integral from a to b of f of x, dx minus the integral from a to b of g of x dx" is false. The correct formula for the integral of a difference is as follows:
∫[a,b] (f(x) - g(x)) dx = ∫[a,b] f(x) dx - ∫[a,b] g(x) dx
In this case, both integrals of f(x) and g(x) must be evaluated separately and then subtracted.
2. The statement "the integral from a to a of f of x, dx equals 2 times the integral from 0 to a of f of x, dx" is false. The integral from a to a does not exist as it represents integration over an empty interval.
3. The statement "the integral from a to b of the product of f of x and g of x, dx equals the integral from a to b of f of x, dx times the integral from a to b of g of x dx" is false. The correct formula for integrating the product of two functions is the following:
∫[a,b] (f(x) * g(x)) dx ≠ ∫[a,b] f(x) dx * ∫[a,b] g(x) dx
The integral of the product of two functions cannot be simplified to the product of their individual integrals.
4. The statement "the integral from a to b of 3 plus f of x, dx equals 3 plus the integral from a to b of f of x, dx" is true. Adding a constant value to a function being integrated does not affect the integral itself. Therefore, the integral of 3 plus f(x) over the interval [a, b] is indeed equal to 3 plus the integral of f(x) over the same interval.
In conclusion, the correct statement is:
~the integral from a to b of 3 plus f of x, dx equals 3 plus the integral from a to b of f of x, dx.
nope. sums and differences work that way, but not products. Consider
f(x) = x
g(x) = x
∫f * ∫g = 1/2 x^2 * 1/2 x^2 = 1/4 x^4
∫fg = ∫x^2 = 1/3 x^3
∫f-g = ∫f - ∫g TRUE
∫[a,a] f(x) dx = 0
NOT 2∫[0,a]
∫3+f = ∫3 + ∫f
NOT 3+∫f