If the integral from 0 to 8 of f(x)=4, then calculate the integral of f(4x) from 0 to 2.
suppose ∫ f(x) dx = F(x)
Then
∫[0,8] f(x) dx = F(8)-F(0) = 4
∫[0,2] f(4x) dx
= ∫[0,2] 1/4 f(4x) d(4x)
= 1/4(F(8)-F(0)) = 1
check:
∫[0,8] x^2 dx = 512/3
∫[0,2] (4x)^2 dx = 16(8/3) = 128/3
Well, well, well, it looks like we have an integral on our hands. Let's dive in, shall we?
First, we're given that the integral from 0 to 8 of f(x) is equal to 4. That's great news - it means f(x) is a nice and constant 4 on that interval.
Now, we want to calculate the integral of f(4x) from 0 to 2. To do that, we need to change our limits of integration. If we let u = 4x, then when x = 0, u = 0 and when x = 2, u = 8. So, our new limits of integration will be from 0 to 8.
Now, let's figure out f(4x). Since we know f(x) is always 4, that means f(4x) will also be 4, no matter what x is. So, we can rewrite the integral as the integral from 0 to 8 of 4 du.
And what do you know? The integral of 4 du is simply 4u. Plugging in our limits of integration, we get 4(8) - 4(0), which simplifies to... drumroll please... 32!
So, the integral of f(4x) from 0 to 2 is equal to 32. Now, that's something to celebrate!
To calculate the integral of f(4x) from 0 to 2, we need to perform a change of variable. Let u = 4x, which implies du = 4dx, and dx = du/4.
We can now rewrite the integral as:
∫[0 to 2] f(4x) dx = ∫[0 to 2] f(u) (du/4)
Since the integral from 0 to 8 of f(x) is known to be 4, we can substitute f(x) with its value and simplify the integral:
∫[0 to 2] f(u) (du/4) = ∫[0 to 2] 4 (du/4)
Notice that the constant 4 is independent of u, so it can be taken out of the integral:
∫[0 to 2] 4 (du/4) = 4 ∫[0 to 2] (du/4)
Integrating (du/4) yields u/4:
4 ∫[0 to 2] (du/4) = 4 * (u/4) *[0 to 2]
Evaluate the integral limits:
4 * (u/4) *[0 to 2] = 4 * (2/4 - 0/4)
Simplify:
4 * (2/4 - 0/4) = 4 * (1/2)
Compute:
4 * (1/2) = 4/2 = 2
Therefore, the integral of f(4x) from 0 to 2 is 2.
To calculate the integral of f(4x) from 0 to 2, you can use the technique of u-substitution. Let's start by setting u = 4x.
Now, we need to find the values of u corresponding to x = 0 and x = 2.
When x = 0, we have u = 4(0) = 0.
When x = 2, we have u = 4(2) = 8.
Next, let's find the differential of u with respect to x. Since u = 4x, we can differentiate both sides with respect to x to get du/dx = 4.
Now, let's rewrite the integral in terms of u:
∫[0 to 2] f(4x) dx = ∫[0 to 2] f(u) * (du/dx) dx
Since du/dx = 4, we can substitute this value back into the integral:
∫[0 to 2] f(u) * (du/dx) dx = ∫[0 to 2] f(u) * 4 dx
We know that the integral from 0 to 8 of f(x) is equal to 4, so we can rewrite the integral using the value of the integral of f(x):
∫[0 to 2] f(u) * 4 dx = 4 * ∫[0 to 2] f(u) dx
Since the integral of f(x) from 0 to 8 is 4, we can substitute this value into the integral:
4 * ∫[0 to 2] f(u) dx = 4 * 4
Therefore, the integral of f(4x) from 0 to 2 is equal to 16.