Find the area of the region enclosed between y=2sin(x) and y=3cos(x) from x=0 to x=0.7π. Hint: Notice that this region consists of two parts.
Oh, I see we have a little math problem here. Well, when it comes to finding areas, I like to think outside the box...or should I say region?
So, let's tackle this in two parts. First, we'll find the area between the x-axis and y=2sin(x). This region is like a sinusoidal roller coaster. Now, the formula to find the area under a curve is to integrate, so let's do some math magic and integrate 2sin(x) from x=0 to x=0.7π.
∫[0,0.7π] (2sin(x)) dx
Now, when you integrate sin(x), you get -cos(x). So, let's plug that in:
[-2cos(x)] from x=0 to x=0.7π
Plug in the limits and we get:
[-2cos(0.7π)] - [-2cos(0)]
But since cos(0) is 1, we simplify it to:
[-2cos(0.7π)] + 2
And there you have it! The area between the x-axis and y=2sin(x) is approximately -2cos(0.7π) + 2.
Now, let's move on to the second part. We want to find the area between the x-axis and y=3cos(x). This region is like a wavy ride on the cosine train. Let's integrate 3cos(x) from x=0 to x=0.7π.
∫[0,0.7π] (3cos(x)) dx
When you integrate cos(x), you get sin(x). So, let's go ahead and plug that in:
[3sin(x)] from x=0 to x=0.7π
Plug in the limits and we get:
[3sin(0.7π)] - [3sin(0)]
But since sin(0) is also 0, we simplify it to:
3sin(0.7π)
And there you have it! The area between the x-axis and y=3cos(x) is approximately 3sin(0.7π).
Now, to find the total area between the two curves, we simply subtract the second part from the first part:
Total Area = (-2cos(0.7π) + 2) - 3sin(0.7π)
And that's the area of the region enclosed between y=2sin(x) and y=3cos(x) from x=0 to x=0.7π. I hope you had as much fun with math as I did!
To find the area of the region enclosed between the two curves, we need to calculate the areas of the two separate parts and then add them together.
Step 1: Determine the points of intersection between the two curves.
Setting the two equations equal to each other:
2sin(x) = 3cos(x)
Divide both sides by 2cos(x):
tan(x) = 3/2
Taking the inverse tangent on both sides:
x = arctan(3/2)
Step 2: Calculate the area of the first part.
To find the area of the region from x = 0 to x = arctan(3/2), we need to integrate the difference between the two curves:
Area of the first part = ∫[0, arctan(3/2)] (3cos(x) - 2sin(x)) dx
Step 3: Calculate the area of the second part.
To find the area of the region from x = arctan(3/2) to x = 0.7π, we need to integrate the difference between the two curves again:
Area of the second part = ∫[arctan(3/2), 0.7π] (2sin(x) - 3cos(x)) dx
Step 4: Add the two areas together to find the total area.
Total area = Area of the first part + Area of the second part
Total area = ∫[0, arctan(3/2)] (3cos(x) - 2sin(x)) dx + ∫[arctan(3/2), 0.7π] (2sin(x) - 3cos(x)) dx
To find the exact value of the area, we need to calculate these integrals using appropriate techniques or a calculator.
To find the area of the region enclosed between two curves, you need to integrate the difference between the two functions over the given interval. In this case, the two curves are y = 2sin(x) and y = 3cos(x).
The first step is to find the x-values at which the two curves intersect. By setting the two equations equal to each other, we have:
2sin(x) = 3cos(x)
Dividing both sides by 2cos(x), we get:
tan(x) = 3/2
Now, we need to find the values of x that satisfy this equation within the given interval [0, 0.7π]. Look for the values of x where the tangent function equals 3/2 within this interval.
Using a calculator, you can find the approximate x-values where tan(x) = 3/2 as x ≈ 0.9828 and x ≈ 2.159.
Now we have two parts of the region: from x = 0 to x ≈ 0.9828, and from x ≈ 0.9828 to x ≈ 2.159.
To find the area of the first part, we integrate the difference between the two curves from x = 0 to x ≈ 0.9828:
Area1 = ∫[0, 0.9828] (2sin(x) - 3cos(x)) dx
Similarly, for the second part, we integrate the difference between the two curves from x ≈ 0.9828 to x ≈ 2.159:
Area2 = ∫[0.9828, 2.159] (2sin(x) - 3cos(x)) dx
Add the two areas together to get the total area of the region:
Total Area = Area1 + Area2
Evaluate these two integrals using integration techniques or numerical methods such as the trapezoidal or Simpson's rule to find the area enclosed by the curves between the given interval.
since 0.7π = 2.199, and
(a) 3cosx > 2sinx for 0 <= x <= 0.983
(b) 2sinx > 3cosx for 0.98279 <= x <= 2.199
The area is
∫[0,0.983] (3cosx-2sinx) dx + ∫[0.983,2.199] (2sinx-3cosx) dx