can you pleaaaaase help me find the area between y=cos(4x) and y=1-cos(4x) 0<=x<=pi/4
i tried to solve it, and i got like : 2*sqrt3+ pi/12
is it right? if not please tell me how can i solve it...i'm really confused
Graph it first. See where the curves meet. If they meet at some angle less than PI/4, then you will have to break up the integration (upper-lower)dx into segments 1->meeting point> otherwise you get a "negative" area which will subtract from your area.
i did all that, i found Pi/12
then , i took the integral from 0 to pi/12, and from pi/12 to pi/4
and i got:
integr from 0 to pi/12 cos(4x)-(1-cos(4x) + integ from pi/12 to pi/4 1-cos(4x)-cos(4x)
but i'm not sure if it was right or not...can u check with me please?
Yes, I can help you find the area between the two curves.
To find the area between two curves, you need to integrate the difference between the upper and lower functions with respect to the given variable. In this case, we will integrate the difference between the functions y = cos(4x) and y = 1 - cos(4x) with respect to x.
To determine the limits of integration, you mentioned that 0 ≤ x ≤ π/4. So we will integrate from x = 0 to x = π/4.
Now, let's calculate the area step by step.
1. First, find the points of intersection between the two curves. Set the two equations equal to each other and solve for x:
cos(4x) = 1 - cos(4x)
2cos(4x) = 1
cos(4x) = 1/2
To find the values of x, where cos(4x) = 1/2, we can use the inverse cosine function:
4x = arccos(1/2)
4x = π/3 or 4x = -π/3
x = π/12 or x = -π/12
Since we are considering the range 0 ≤ x ≤ π/4, the only relevant solution for x is x = π/12.
2. Now, we can determine the upper and lower functions within the given limits.
Let's substitute x = π/12 into both equations:
y = cos(4(π/12)) = cos(π/3) = 1/2
y = 1 - cos(4(π/12)) = 1 - cos(π/3) = 1 - 1/2 = 1/2
So the upper function is y = 1/2 and the lower function is y = 1/2.
3. Calculate the area by taking the integral of the difference between the upper and lower functions:
Area = ∫[a,b] (upper function - lower function) dx
= ∫[0,π/4] (1/2 - 1/2) dx
= ∫[0,π/4] 0 dx
= 0
Therefore, the area between the curves y = cos(4x) and y = 1 - cos(4x) within the range 0 ≤ x ≤ π/4 is 0.
Based on your calculations, you mentioned getting 2√3 + π/12, which appears to be incorrect, as the area between the curves is zero.