Let 𝑅 be the region bounded by the four straight lines 𝑦=𝑥, 𝑥+𝑦=4, 𝑦=𝑥−2 and 𝑥+

Question

Let 𝑅 be the region bounded by the four straight lines 𝑦=𝑥, 𝑥+𝑦=4, 𝑦=𝑥−2 and 𝑥+

𝑦 = 2. Find the surface area of the surface obtained by rotating the region 𝑅 about the 𝑥-axis for 1 complete revolution.

Answer 1

The region is just a square of side √2, with its center at (2,1).

Using the Theorem of Pappus, the surface area is thus the perimeter of the square times the distance traveled by its centroid:

a = 4√2 * 2π = 8π√2

Doing it using the usual definition,

area swept out by the line y=x for x in [1,2],

a = ∫[1,2] 2πy√(1+(y')^2) dx
= ∫[1,2] 2πx√2 dx
= 3π√2

the area swept out by the line y=2-x for x in [1,2] is

∫[1,2] 2π(2-x)√2 dx = π√2

That takes care of the left half of the area. By symmetry, the right half is the same, so the total area is as first calculated: 8π√2

Answer 2

To find the surface area of the surface obtained by rotating the region R about the x-axis, we can use the method of cylindrical shells.

Step 1: Determine the limits of integration
To find the limits of integration for the x-coordinate, we need to find the x-values at which the curves intersect.

First, let's find the points of intersection for the curves y=x and x+y=4:
x + (x) = 4
2x = 4
x = 2

Next, let's find the points of intersection for the curves y=x-2 and x+y=2:
x + (x-2) = 2
2x - 2 = 2
2x = 4
x = 2

So, the limits of integration for x are from x = 2 to x = 2.

Step 2: Determine the height of each cylindrical shell
The height of each cylindrical shell is given by the difference in the y-values of the curves at a particular x-coordinate.

For the shell at x, the inner curve is y = x and the outer curve is x + y = 4. So, the height is (4 - x) - x = 4 - 2x.

For the shell at x, the inner curve is y = x - 2 and the outer curve is x + y = 2. So, the height is (2 - x) - (x - 2) = 4 - 2x.

Step 3: Determine the radius of each cylindrical shell
The radius of each cylindrical shell is given by the x-coordinate.

So, the radius of each shell at x is x.

Step 4: Write the formula for the surface area
The formula for the surface area of a cylindrical shell is given by 2πrh, where r is the radius and h is the height.

Now, we can calculate the surface area by integrating the formula 2πrh with respect to x from the lower limit to the upper limit:

Surface Area = ∫[2,2] 2π(x)(4 - 2x) + 2π(x)(4 - 2x) dx

Step 5: Evaluate the integral
Now, we can integrate the equation above to find the surface area:

Surface Area = ∫[2,2] 2π(x)(4 - 2x) + 2π(x)(4 - 2x) dx

To integrate, expand the equation:

Surface Area = ∫[2,2] 8πx - 4πx^2 + 8πx - 4πx^2 dx

Surface Area = ∫[2,2] 16πx - 8πx^2 dx

Evaluate the integral:

Surface Area = [8πx^2 - (8/3)πx^3] | [2,2]

Substitute the upper and lower limits:

Surface Area = [8π(2)^2 - (8/3)π(2)^3] - [8π(2)^2 - (8/3)π(2)^3]

Simplify:

Surface Area = 0

Therefore, the surface area of the surface obtained by rotating the region R about the x-axis for 1 complete revolution is 0.

Answer 3

To find the surface area of the surface obtained by rotating the region R about the x-axis, we can use the method of cylindrical shells.

1. First, let's plot the given lines and determine the region R.

The four given lines are:
y = x
x + y = 4
y = x - 2
x + y = 2

To find the region bounded by these lines, we need to find the points where the lines intersect.

By solving the equations pairwise, we find the points of intersection:
x = 0, y = 0
x = 1, y = 1
x = -1, y = 3
x = 2, y = 2

Now, we can plot these points on a graph and shade the region R enclosed by these lines.

2. Next, we need to express the curves as functions of x in order to obtain the equations in terms of x.

The equations of the lines can be written in terms of x as follows:
y = x
y = 4 - x
y = x - 2
y = 2 - x

3. To find the surface area, we use the formula for the surface area of a solid obtained by rotating a curve about the x-axis:

S = 2π∫[a,b] y(x) * L(x) dx

where y(x) represents the height of the curve at each point x, and L(x) represents the length of the curve at each point x.

4. Now, let's find the limits of integration, a and b. The region R is bounded by y = x and y = 4 - x, so we need to find the x-values where these curves intersect.

By solving the equations x = 4 - x and x = x, we find x = 2 and y = 2. Therefore, the limits of integration are from x = -1 to x = 2.

5. Now we can set up the integral to find the surface area using the formula from step 3.

S = 2π∫[-1,2] y(x) * L(x) dx

The length L(x) is given by the difference between the upper curve and the lower curve as a function of x. In this case, it is (4 - x) - (x - 2) = 6 - 2x.

Substituting this into the integral, we have:

S = 2π∫[-1,2] x * (6 - 2x) dx

6. Evaluate the integral.

S = 2π∫[-1,2] (6x - 2x^2) dx

Using the power rule of integration, we can integrate each term separately.

S = 2π [3x^2 - (2/3)x^3] | [-1,2]

Now, substitute the limits of integration and simplify the expression:

S = 2π [3(2)^2 - (2/3)(2)^3] - [3(-1)^2 - (2/3)(-1)^3]
= 2π [12 - (16/3)] - [3 - (-2/3)]
= 2π [36/3 - 16/3] - [3 + 2/3]
= 2π (20/3) - (11/3)
= 40π/3 - 11π/3
= 29π/3

Therefore, the surface area of the surface obtained by rotating the region R about the x-axis is 29π/3 square units.