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:
- When y = 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.
