Suppose the integral from 2 to 6 of g of x, dx equals 12 and the integral from 5 to 6 of g of x, dx equals negative 3 , find the value of the integral from 2 to 5 of 3 times g of x, dx.
A. 45 <---- My choice
B. 15
C. 5
D. 3
My work:
INT g(x)dx from 2 to 6=
INT g(x)dx from 2 to 5 + INT(g(x) from 5 to 6
12=Int(g(x) from 2 to 5 -3
so INT gIx)dx from 2 to 5 = 15
so three times int(g)x from 2 to 5 is then 3*15
Am I correct?
you are correct
Well, let me put on my clown wig and analyze this for you!
You're on the right track! The integral from 2 to 6 of g(x)dx can be broken down into two parts: the integral from 2 to 5 and the integral from 5 to 6. You correctly set up the equation:
12 = int(g(x) from 2 to 5 - 3
If we solve for the integral from 2 to 5, we get:
int(g(x) from 2 to 5 = 15
Now, you want to find the value of the integral from 2 to 5 of 3 times g(x)dx. Since the integral is a linear operator, we can pull out the constant factor:
3 * int(g(x) from 2 to 5 = 3 * 15
And guess what? That equals 45, my friend! So, the correct answer is A. 45.
Keep up the good work and keep those clown questions coming!
Yes, you are correct. The integral of 3 times g(x) from 2 to 5 is indeed 45.
By using the linearity property of integrals, we can rewrite the integral from 2 to 6 of g(x)dx as the sum of the integral from 2 to 5 of g(x)dx and the integral from 5 to 6 of g(x)dx.
12 = integral from 2 to 5 of g(x)dx + (-3)
Rearranging this equation, we find that the integral from 2 to 5 of g(x)dx is 12 + 3 = 15.
Then, multiplying this by 3 gives us the value of the integral from 2 to 5 of 3 times g(x)dx, which is 3 * 15 = 45. Therefore, the correct choice is A. 45.
Yes, you are correct. Here's how you can arrive at the correct answer:
Given that the integral from 2 to 6 of g(x) dx equals 12 and the integral from 5 to 6 of g(x) dx equals -3, you can find the value of the integral from 2 to 5 of 3 times g(x) dx.
Using the linearity property of integrals, you can split the integral from 2 to 6 into two separate integrals: one from 2 to 5 and the other from 5 to 6.
So, the integral from 2 to 6 of g(x) dx can be written as:
∫[2,6] g(x) dx = ∫[2,5] g(x) dx + ∫[5,6] g(x) dx
Given that the integral from 2 to 6 of g(x) dx equals 12 and the integral from 5 to 6 of g(x) dx equals -3, you can rewrite the equation as:
12 = ∫[2,5] g(x) dx + (-3)
Rearranging the equation, you get:
∫[2,5] g(x) dx = 12 - (-3) = 15
Now, you need to find the value of the integral from 2 to 5 of 3 times g(x) dx. Using the linearity property of integrals again, you can factor out the constant 3 from the integral:
∫[2,5] 3 * g(x) dx = 3 * ∫[2,5] g(x) dx
Substituting the value of ∫[2,5] g(x) dx as 15, you get:
3 * 15 = 45
Therefore, the value of the integral from 2 to 5 of 3 times g(x) dx is 45. So, the correct answer is A. 45.