2. Sketch the region in the first quadrant that is bounded by the graphs of y=x^3, y=4X , and 2x+y-3=0 that lies below both straight lines. Find the area of this region.
I can see the graph in front of me. which function do I subtract from which?
To determine which function to subtract from which, we need to find the points of intersection between the curves.
First, let's find the points of intersection between the curve y = x^3 and the line y = 4x:
Setting x^3 = 4x and rearranging the equation, we get:
x^3 - 4x = 0
Factoring out an x, we have:
x(x^2 - 4) = 0
Solving for x, we find three solutions:
x = 0, x = -2, and x = 2
Next, we find the points of intersection between the curve y = x^3 and the line 2x + y - 3 = 0:
Substituting y = x^3 into the equation of the line, we get:
2x + x^3 - 3 = 0
Simplifying, we have:
x^3 + 2x - 3 = 0
Unfortunately, this equation does not have a simple algebraic solution, so we need to use numerical methods or graphical approximation to find the solution. Through numerical methods or graphic approximation, we find the approximate values: x ≈ -1.75, -0.65, and 1.39.
Now that we have the points of intersection between the curves, we can determine which function is above the other in different regions.
In the first region, between x = -2 and x = -1.75, the curve y = x^3 is above the line 2x + y - 3 = 0. Therefore, we will subtract the equation of the line from the curve in this region.
In the second region, between x = -1.75 and x = -0.65, the line 2x + y - 3 = 0 is above the curve y = x^3. Thus, we will subtract the curve from the equation of the line in this region.
In the third region, between x = -0.65 and x = 0, the curve y = x^3 is above the line 2x + y - 3 = 0. So, we subtract the equation of the line from the curve again.
In the fourth region, between x = 0 and x = 1.39, the line 2x + y - 3 = 0 is above the curve y = x^3. Thus, we subtract the curve from the equation of the line once again.
Finally, in the fifth region, between x = 1.39 and x = 2, the curve y = x^3 is above the line 2x + y - 3 = 0. Hence, we subtract the equation of the line from the curve in this region.
By determining which function is above the other in each region, you can now subtract the equation of the line from the curve or vice versa to find the respective regions. Once you have the regions specified, you can integrate each region separately to find the area of the bounded region in the first quadrant.