Find the area between y=3sinx and y=3cosx over the interval [0,pi].
To find the area between the two curves y=3sinx and y=3cosx over the interval [0,pi], we can use integration.
First, let's find the points of intersection between the two curves.
Setting y=3sinx equal to y=3cosx, we have:
3sinx = 3cosx
Dividing both sides by 3, we get:
sinx = cosx
To find the values of x that satisfy this equation, we can use the identity sinx = cos(π/2 - x).
Therefore, sinx = cos(π/2 - x) implies that:
x = π/2 - x
Adding x to both sides, we obtain:
2x = π/2
Dividing both sides by 2, we have:
x = π/4
So, the points of intersection between the two curves are (π/4, 3/√2) and (π/4, 3/√2).
Now, let's integrate to find the area between the curves. We will integrate the absolute difference between the two curves over the interval [0, π].
The area A can be calculated as follows:
A = ∫[0,π] |(3sinx - 3cosx)| dx
Splitting the integral into two parts, we need to consider the values of sinx - cosx in the intervals [0,π/4] and [π/4,π].
For the interval [0,π/4], sinx is greater than cosx, so the absolute difference between the two is:
3sinx - 3cosx
For the interval [π/4,π], cosx is greater than sinx, so the absolute difference between the two is:
3cosx - 3sinx
Therefore, the integral A can be written as:
A = ∫[0,π/4] (3sinx - 3cosx) dx + ∫[π/4,π] (3cosx - 3sinx) dx
Evaluating these integrals, we get:
A = [-3cosx - 3sinx] | [0,π/4] + [-3sinx - 3cosx] | [π/4,π]
Evaluating these limits, we have:
A = [-3cos(π/4) - 3sin(π/4)] - [-3cos(0) - 3sin(0)] + [-3sin(π) - 3cos(π)] - [-3sin(π/4) - 3cos(π/4)]
Calculating these values, we get:
A = [-(3/√2) - (3/√2)] - [(-3) - 0] + [0 - (-3)] - [(-3/√2) - (-3/√2)]
Simplifying further:
A = -6/√2 + 3 + 3 + 6/√2
Combining similar terms:
A = 6 + 6/√2
Therefore, the area between the curves y=3sinx and y=3cosx over the interval [0,π] is 6 + 6/√2 square units.
To find the area between two curves, we need to find the definite integral of the difference between the upper curve and the lower curve over the given interval.
In this case, the upper curve is y = 3sin(x) and the lower curve is y = 3cos(x). We want to find the area between these two curves over the interval [0, π].
Step 1: Find the points of intersection between the two curves.
To find the points of intersection, we set the two equations equal to each other and solve for x:
3sin(x) = 3cos(x)
Dividing both sides by 3:
sin(x) = cos(x)
Using the identity sin(x) = cos(π/2 - x), we can rewrite the equation as:
sin(x) = sin(π/2 - x)
This implies:
x = π/2 - x + 2nπ, where n is an integer
Simplifying, we get:
2x = π/2 + 2nπ
x = π/4 + nπ
To find the intersection points within the interval [0, π], we plug in n = 0 and n = 1:
For n = 0: x = π/4
For n = 1: x = 5π/4
Step 2: Set up the integral.
The area between the curves is given by the integral of the difference between the upper curve and the lower curve over the interval [0, π]:
Area = ∫[0,π] (3sin(x) - 3cos(x)) dx
Step 3: Evaluate the integral.
Evaluating this integral:
Area = ∫[0,π] (3sin(x) - 3cos(x)) dx
Using the identities ∫ sin(x) dx = -cos(x) and ∫ cos(x) dx = sin(x), we can rewrite the integral as:
Area = [-3cos(x) - 3sin(x)] evaluated from 0 to π
Evaluating at the upper and lower limits:
Area = (-3cos(π) - 3sin(π)) - (-3cos(0) - 3sin(0))
Since cos(π) = -1, sin(π) = 0, cos(0) = 1, and sin(0) = 0, we can simplify:
Area = (-3(-1)) - (-3(1)) = 3 - (-3) = 6
Therefore, the area between the curves y = 3sin(x) and y = 3cos(x) over the interval [0, π] is 6 square units.
Find the indefinite integral of
3 sin x - 3 cos x.
It is -3 sin x - 3 cos x.
Finally, evaluate that at x = pi and subtract the value at x = 0, the two limts of integration.
I get (0 + 3) - (0 -3(1)) = 6
Over part of the interval, the 3 cos x curve is higher than 3 sin x. That part of the area integral is counted as negative
As drwls noted, the cosine curve is above the sine curve from x =0 to x=pi/4 and from pi/4 to pi, the sine curve is higher.
So the calculation has to be done in two parts
Area = integral(3cosx - 3sinx)dx from 0 to pi/4 + integral(3sinx - 3cosx)dx from pi/4 to pi.
I got 6√2 as a final answer