6. the regions bounded by the graphs of ( y = 6x ) and ( y = 5x^{2}-x^{3} ) are shaded in the figure above…

6. the regions bounded by the graphs of ( y = 6x ) and ( y = 5x^{2}-x^{3} ) are shaded in the figure above. the graphs intersect at ( x = 0,x = 2 ), and ( x = 3 ). which of the following gives the sum of the areas of the shaded regions? (a) ( int_{0}^{3}(6x-(5x^{2}-x^{3}))dx ) (b) ( int_{0}^{2}(6x-(5x^{2}-x^{3}))dx+int_{2}^{3}((5x^{2}-x^{3})-6x)dx ) (c) ( int_{0}^{2}(5x^{2}-x^{3})dx+int_{2}^{3}6xdx ) (d) ( int_{0}^{2}6xdx+int_{2}^{3}(5x^{2}-x^{3})dx )

6. the regions bounded by the graphs of ( y = 6x ) and ( y = 5x^{2}-x^{3} ) are shaded in the figure above. the graphs intersect at ( x = 0,x = 2 ), and ( x = 3 ). which of the following gives the sum of the areas of the shaded regions? (a) ( int_{0}^{3}(6x-(5x^{2}-x^{3}))dx ) (b) ( int_{0}^{2}(6x-(5x^{2}-x^{3}))dx+int_{2}^{3}((5x^{2}-x^{3})-6x)dx ) (c) ( int_{0}^{2}(5x^{2}-x^{3})dx+int_{2}^{3}6xdx ) (d) ( int_{0}^{2}6xdx+int_{2}^{3}(5x^{2}-x^{3})dx )

Answer

Explanation:

Step1: Determine the upper and lower functions in different intervals

For the interval (0\leq x\leq2), we compare (y = 6x) and (y=5x^{2}-x^{3}). Let (f(x)=6x-(5x^{2}-x^{3})=x^{3}-5x^{2}+6x=x(x - 2)(x - 3)). When (x\in[0,2]), (f(x)\geq0), so (y = 6x) is the upper - function and (y=5x^{2}-x^{3}) is the lower - function. The area of the region in the interval ([0,2]) is given by (\int_{0}^{2}\left[6x-(5x^{2}-x^{3})\right]dx).

Step2: Determine the upper and lower functions in the other interval

For the interval (2\leq x\leq3), we again consider (f(x)=6x-(5x^{2}-x^{3})). When (x\in[2,3]), (f(x)\leq0), so (y = 5x^{2}-x^{3}) is the upper - function and (y = 6x) is the lower - function. The area of the region in the interval ([2,3]) is given by (\int_{2}^{3}\left[(5x^{2}-x^{3})-6x\right]dx).

Step3: Calculate the total area

The sum of the areas of the shaded regions is the sum of the areas of the two sub - regions. So the total area (A=\int_{0}^{2}(6x-(5x^{2}-x^{3}))dx+\int_{2}^{3}((5x^{2}-x^{3})-6x)dx).

Answer:

B. (\int_{0}^{2}(6x-(5x^{2}-x^{3}))dx+\int_{2}^{3}((5x^{2}-x^{3})-6x)dx)