Question: Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration.
The correct answer is –
To sketch the region, we start by plotting the lines on a coordinate axis.
First, we can find the x-intercept of the line 2x + y = 8 by setting y = 0:
2x + 0 = 8
x = 4
So, the line passes through the point (4, 0).
Next, we can find the points where the line intersects the other two given lines:
- When y = 2:
2x + y = 8
2x + 2 = 8
x = 3
So, the line passes through the point (3, 2).
- When y = 4:
2x + y = 8
2x + 4 = 8
x = 2
So, the line passes through the point (2, 4).
We can now plot the lines and shade the region bounded by them and the y-axis:
|
4 +————–+
| |
3 +————–+
| Region |
2 +——+ |
| | |
1 +——+——-+
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
+–+—+—+—+
0 2 3 4
To find the area of this region, we can integrate the area of the vertical strips that make up the region.
The strips are bounded by the y-axis on one side and the line 2x + y = 8 on the other side. We can express this line in terms of x as:
y = 8 – 2x
So the height of each strip is given by the difference between the y-coordinate of the line and the y-coordinate of the y-axis (which is 0).
The width of each strip is dx.
Therefore, the area of each strip is:
dA = (8 – 2x) dx
To find the total area, we integrate this expression over the range of x values that define the region:
A = ∫(from x=0 to x=2) (8 – 2x) dx + ∫(from x=2 to x=3) (4 – 2x) dx + ∫(from x=3 to x=4) (2) dx
Simplifying this expression, we get:
A = [8x – x^2] from x=0 to x=2 + [4x – x^2] from x=2 to x=3 + [2x] from x=3 to x=4
A = 12
Therefore, the area of the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis is 12 square units.
