Suppose the integral from 2 to 8 of g of x, dx equals 5, and the integral from 6 to 8 of g of x, dx equals negative 3, find the value of the integral from 2 to 6 of 2 times g of x, dx .
8 MY ANSWER
12
16
4
Hmmm. Let's trudge through those words. We have
∫[2,8] g(x) dx = 5
∫[6,8] g(x) dx = -3
we know that
∫[2,6] + ∫[6,8] = ∫[2,8]
∫[2,6] + (-3) = 5
∫[2,6] g(x) = 8
∫[2,6] 2g(x) = 2*8 = 16
Forgot that 2, eh?
Well, 8 is certainly a number, but I'm afraid it's not the correct answer in this case. Let's break it down:
We know that the integral from 2 to 8 of g(x) dx equals 5 and the integral from 6 to 8 of g(x) dx equals -3.
Now, we want to find the value of the integral from 2 to 6 of 2 times g(x) dx.
Using linearity of integration, we can split up this integral as follows:
∫(2g(x)) dx = 2∫g(x) dx
Since we're given that ∫g(x) dx from 2 to 8 equals 5, we can substitute it in:
2∫g(x) dx = 2 * 5 = 10
So, the value of the integral from 2 to 6 of 2g(x) dx is equal to 10, which is not one of the options you provided. Therefore, none of the given options are correct.
To find the value of the integral from 2 to 6 of 2 times g(x), dx, we can use the linearity property of integrals. The linearity property states that the integral of a constant times a function is equal to the constant times the integral of the function.
So, we can rewrite the integral as follows:
∫[2 to 6] 2 * g(x) dx = 2 * ∫[2 to 6] g(x) dx
Since we are given that the integral from 2 to 8 of g(x) is equal to 5 and the integral from 6 to 8 of g(x) is equal to -3, we can calculate the value of the integral from 2 to 6 of g(x) by subtracting the integral from 6 to 8 of g(x) from the integral from 2 to 8 of g(x).
∫[2 to 6] g(x) dx = ∫[2 to 8] g(x) dx - ∫[6 to 8] g(x) dx
Plugging in the given values, we get:
∫[2 to 6] g(x) dx = 5 - (-3)
∫[2 to 6] g(x) dx = 5 + 3
∫[2 to 6] g(x) dx = 8
Finally, we can substitute this value back into the original expression to find the integral from 2 to 6 of 2 times g(x):
∫[2 to 6] 2 * g(x) dx = 2 * 8
∫[2 to 6] 2 * g(x) dx = 16
Therefore, the value of the integral from 2 to 6 of 2 times g(x) is 16. Thus, the correct answer is 16.
To find the value of the integral from 2 to 6 of 2 times g(x), dx, we can use the properties of integrals.
First, let's rewrite the given information in terms of the integral notation:
∫[2 to 8] g(x) dx = 5
∫[6 to 8] g(x) dx = -3
Now, we can move on to the integral we need to solve. Let's consider the integral from 2 to 6 of 2 times g(x), dx:
∫[2 to 6] 2g(x) dx
Since the integrals from 2 to 8 and from 6 to 8 are given, let's split the integral into two parts:
∫[2 to 6] 2g(x) dx = ∫[2 to 6] 2g(x) dx + ∫[6 to 8] 2g(x) dx
Using the linearity property of integrals, we can move the constant factor of 2 outside the integrals:
∫[2 to 6] 2g(x) dx = 2 ∫[2 to 6] g(x) dx + 2 ∫[6 to 8] g(x) dx
Now, we can substitute the given values:
2 ∫[2 to 6] g(x) dx + 2 ∫[6 to 8] g(x) dx = 2(5) + 2(-3)
Simplifying this expression:
2(5) + 2(-3) = 10 - 6 = 4
Therefore, the value of the integral from 2 to 6 of 2 times g(x), dx is 4.
So, the correct answer is 4.