Find the area bounded by the curves y=sin(4x) and y=0 for x between x=0 and x=3pi/8.
Notice the period is 2π/4 or π/2 , (90°)
so your region is 1/2 a sine curve from 0 to π/4, plus a bit of the sine below the x-axis,
so take it in two parts,
from x = 0 to π/4 then the small bit below
don't forget to take
∫sin4x dx from 0 to π/4 + ∫-sin4x from π/4 to 3π/8
To find the area bounded by the curves y = sin(4x) and y = 0 for x between x = 0 and x = 3π/8, we can use integration.
Step 1: Find the x-values where the two curves intersect.
Set sin(4x) = 0:
0 = sin(4x)
2πn = 4x
x = πn/2 for n = 0, 1, 2, 3...
Since we are interested in the x-values between x = 0 and x = 3π/8, we can see that the curves intersect at x = 0 and x = π/8.
Step 2: Set up the integral to find the area using the formula:
A = ∫[a, b] (f(x) - g(x)) dx
where f(x) is the upper curve and g(x) is the lower curve.
In this case, f(x) = sin(4x) and g(x) = 0. Since y = 0 is the x-axis, it will serve as the lower curve.
So the area A is given by:
A = ∫[0, 3π/8] (sin(4x) - 0) dx
Step 3: Evaluate the integral:
A = ∫[0, 3π/8] sin(4x) dx
We can solve this by using a u-substitution. Let u = 4x, then du = 4 dx.
Our integral then becomes:
A = (1/4) ∫[0, 3π/2] sin(u) du
A = (1/4) [-cos(u)] from 0 to 3π/8
A = (1/4) [-cos(3π/8) - (-cos(0))]
A = (1/4) [-cos(3π/8) + 1]
Step 4: Calculate the final result:
A ≈ (1/4) * (0.9239 + 1)
A ≈ 0.23098
Therefore, the area bounded by the curves y = sin(4x) and y = 0 for x between x = 0 and x = 3π/8 is approximately 0.231 square units.
To find the area bounded by the curves y = sin(4x) and y = 0, we need to find the definite integral of the difference between the two curves over the given interval.
Step 1: Graph the curves
First, start by graphing the two curves on a coordinate system. The curve y = sin(4x) is a periodic function with a period of π/2, so it completes four full cycles in the interval from x = 0 to x = 3π/8. The curve y = 0 is simply the x-axis.
Step 2: Identify the interval
Next, identify the x-values that bound the region of interest. In this case, we are interested in the area between x = 0 and x = 3π/8.
Step 3: Set up the integral
To find the area, we will set up the following definite integral:
A = ∫[from 0 to 3π/8] (sin(4x) - 0) dx
Step 4: Evaluate the integral
Integrating sin(4x) with respect to x gives us:
A = ∫[from 0 to 3π/8] sin(4x) dx = (-1/4) * cos(4x) + C
Evaluate the integral using the limits of integration:
A = [(-1/4) * cos(4(3π/8))] - [(-1/4) * cos(4(0))]
Simplifying this further:
A = (-1/4) * cos(3π/2) - (-1/4) * cos(0)
A = (-1/4) * 0 - (-1/4) * 1
A = 1/4
Therefore, the area bounded by the curves y = sin(4x) and y = 0, for x between x = 0 and x = 3π/8, is 1/4 square units.