the area of the first quadrant region bounded by the y-axis, the line y=4-x and the graph of y=x-cosx is approximately: a) 4.50 square units, b) 4.54 square units, c) 4.56 square units, d) 4.58 square units, e) 5.00 square units
To find the area of the first quadrant region bounded by the y-axis, the line y=4-x, and the graph of y=x-cos(x), we need to set up an integral and evaluate it.
First, we need to find the x-values where the line y=4-x intersects the graph of y=x-cos(x). To do this, set the two equations equal to each other:
4 - x = x - cos(x)
Rearranging the equation:
2x - cos(x) = 4
To solve for x, we'll use numerical methods such as Newton's method or a graphing calculator. After solving, we find two values of x: x ≈ 1.097 and x ≈ 3.026.
The area of the region between the y-axis and the line y=4-x can be found using the definite integral:
∫[0,a] (4 - x) dx,
where a is the x-value where the line intersects the y=x-cos(x) graph (approximately x ≈ 1.097).
Evaluating this integral gives us the area of the region between the y-axis and the line y=4-x:
∫[0,1.097] (4 - x) dx ≈ 4.565
Next, we need to find the area between the line y=4-x and the graph of y=x-cos(x) in the same interval from x=1.097 to x=3.026. We can find this area using another definite integral:
∫[1.097,3.026] (4 - x - (x - cos(x))) dx.
Evaluating this integral gives us the area of the region between the line y=4-x and the graph of y=x-cos(x):
∫[1.097,3.026] (4 - x - (x - cos(x))) dx ≈ 0.015
Finally, to find the total area of the first quadrant region, we add the two areas together:
Total Area ≈ 4.565 + 0.015 ≈ 4.58
Therefore, the area is approximately 4.58 square units. Thus, the correct answer is d) 4.58 square units.