The discussion revolves around finding the area of shaded regions defined by specific curves in two separate problems. The first problem involves the curves y = x^3 + x^2 - 6x and y = 6x, while the second problem involves y = -x^4 and y = x^4 - 32.
Some participants have provided insights into the areas to be calculated and the corresponding integrals. There is an acknowledgment of a minor correction in the notation, but no explicit consensus has been reached on the final answers.
The original poster requested a step-by-step process for solving the problems, indicating a need for guidance rather than direct solutions. The thread has been moved to a specific homework help category.
HallsofIvy said:For (1) the shaded region lies below the graph of [itex]y= x^3+ x^2- 6x[/itex] and above the graph of [itex]y= 6x[/itex] for x between -4 and 0. Its area is given by
[tex]\int_{-4}^0 [(x^3+ x^2- 6x)- 6x]dx[/tex]
For (2) the shaded region lies below the graph of [itex]y= -x^4[/itex] and above the graph of [itex]y= x^4- 32[/itex]. Those cross where [itex]y= -x^4= x^4- 32[/itex] or [itex]2x^4= 32[/itex] so that [itex]x^4= 16[/itex]: x= -2 and x= 2. Its area is given by
[tex]\int_{-2}^2 [-x^4- (x^4- 32)]dx[/tex]