Suppose the integral from 2 to 10 of g of x, dx equals 10 and the integral from 8 to 10 of g of x, dx equals negative 6, find the value of the integral from 2 to 8 of one-half times g of x, dx .

∫[2,8] 1/2 g(x) dx = 1/2 ∫[2,8] g(x) dx

Now, assuming all integrals are of g(x), we have
∫[2,10] = ∫[2,8] + ∫[8,10]
10 = ∫[2,8] - 6
∫[2,8] = 16
1/2 ∫[2,8] = 8

Well, well, well, looks like we have some integral gymnastics going on here. Let's break it down and solve it with a touch of humor, shall we?

First, we'll start with the given information: the integral from 2 to 10 of g(x) dx equals 10. This tells us that when we integrate g(x) from 2 to 10, we end up with a value of 10. Nothing too tricky so far.

Next up, we have the integral from 8 to 10 of g(x) dx equals -6. This means that when we integrate g(x) from 8 to 10, we get a value of -6. Okay, things are starting to get interesting.

Now, we want to find the value of the integral from 2 to 8 of 1/2 * g(x) dx. Since we're multiplying g(x) by 1/2, we can think of this as shrinking the function down a bit. It's like taking a big cake and cutting it in half. Yummy!

To find the value of this integral, we can break it up into two parts: the integral from 2 to 8 of g(x) dx, and the integral from 2 to 8 of 1/2 dx. The second part is easy peasy lemon squeezy – it's just the integral of 1/2, which we can simplify to (1/2) * (8 - 2) = 3.

Now, let's focus on the first part: the integral from 2 to 8 of g(x) dx. We know that the integral from 2 to 10 of g(x) dx is 10, and the integral from 8 to 10 of g(x) dx is -6. So, if we subtract the second integral from the first, we get 10 - (-6) = 16.

Putting it all together, the value of the integral from 2 to 8 of 1/2 * g(x) dx is 3 + 16 = 19. Ta-da! We're done, my friend. Time to celebrate with some integral-shaped party hats!

To find the value of the integral from 2 to 8 of one-half times g of x, dx, we can use the properties of integrals.

Given:
∫2 to 10 g(x) dx = 10
∫8 to 10 g(x) dx = -6

Now, let's consider the integral from 2 to 10 of one-half times g of x, dx:
∫2 to 10 (1/2) * g(x) dx

We can rewrite this as:
(1/2) * ∫2 to 10 g(x) dx

Using the first given information, we know that ∫2 to 10 g(x) dx = 10. Substituting this value into the equation, we have:
(1/2) * 10

Now, let's solve for the value:
(1/2) * 10 = 5

Therefore, the value of the integral from 2 to 8 of one-half times g of x, dx is 5.

To find the value of the integral from 2 to 8 of one-half times g(x), dx, we can use the properties of integrals and apply them to the given information.

Let's break down the problem step by step:

1. The first piece of information we have is that the integral from 2 to 10 of g(x), dx equals 10. This can be written as:

∫(2 to 10) g(x) dx = 10

2. The second piece of information is that the integral from 8 to 10 of g(x), dx equals -6. This can be written as:

∫(8 to 10) g(x) dx = -6

Now, our goal is to find the value of the integral from 2 to 8 of one-half times g(x), dx. We can rewrite it as:

∫(2 to 8) (1/2)g(x) dx.

Since we are multiplying the function g(x) by 1/2, it means that the area under the curve will be half compared to the original function.

Now, let's use the properties of integrals to find the value:

∫(2 to 8) (1/2)g(x) dx = (1/2) * (∫(2 to 8) g(x) dx)

Here, we have factored out 1/2 from the integral because of the property that the integral of a constant times a function is equal to the constant times the integral of the function.

Now, let's substitute the known values of the integrals we have:

∫(2 to 8) (1/2)g(x) dx = (1/2) * (∫(2 to 10) g(x) dx - ∫(8 to 10) g(x) dx)

Plugging in the values:

∫(2 to 8) (1/2)g(x) dx = (1/2) * (10 - (-6))
= (1/2) * (10 + 6)
= (1/2) * 16
= 8

Therefore, the value of the integral from 2 to 8 of one-half times g(x), dx is 8.