Write limits of integration for the integral \int_W f(x,y,z)\,dV, where W is the quarter cylinder shown, if the length of the cylinder is 1 and its radius is 1.
We don't see the figure of the cylinder.
But if it is a quarter of the cylinder, no need for integration, use
V=πr²h.
To determine the limits of integration for the given integral over the quarter cylinder, we need to determine the range of values for each variable (x, y, z) within the region W.
First, let's consider the height of the cylinder (z-coordinate). Since the length of the cylinder is given as 1, the range for z would be from 0 to 1.
Next, we need to consider the x and y-coordinates. Looking at the quarter cylinder, we can see that x ranges from 0 to 1, and y ranges from 0 to √(1 - x^2).
As a result, the limits of integration for the given integral would be:
- For z: 0 to 1
- For x: 0 to 1
- For y: 0 to √(1 - x^2)
Therefore, the integral can be written as:
∫∫∫_W f(x,y,z) dV = ∫[0,1] ∫[0,√(1 - x^2)] ∫[0,1] f(x,y,z) dz dy dx
To determine the limits of integration for the given integral, we need to think about the region of space represented by the quarter cylinder.
First, let's consider the cylindrical coordinate system since it is particularly suitable for this shape.
In cylindrical coordinates, we have:
x = r * cos(theta)
y = r * sin(theta)
z = z
Now, let's analyze the quarter cylinder's boundaries:
1. The length of the cylinder is given as 1. Since the quarter cylinder only covers a fourth of the entire length, the z-coordinate should range from 0 to 1/4.
0 ≤ z ≤ 1/4
2. The radius of the cylinder is given as 1. Since the quarter cylinder only covers a quarter of the entire cross-sectional area, the radial coordinate should go from 0 to 1.
0 ≤ r ≤ 1
3. The angle θ represents the azimuthal angle, and since the quarter cylinder covers only a quarter of the entire azimuthal angle, we restrict θ to a range of 0 to π/2.
0 ≤ θ ≤ π/2
Combining all the limits:
0 ≤ z ≤ 1/4
0 ≤ r ≤ 1
0 ≤ θ ≤ π/2
Therefore, the limits of integration for the given integral \int_W f(x,y,z)\,dV over the quarter cylinder shown are:
∫(0 to π/2) ∫(0 to 1) ∫(0 to 1/4) f(r*cos(theta), r*sin(theta), z) * r dz dr d(theta)