(A) Explanation Theorem used: If f and g are continuous and f ( x ) ≥ g ( x ) over the interval [ a , b ] , then the area bounded by y = f ( x ) and y = g ( x ) for a ≤ x ≤ b is given exactly by A = ∫ a b [ f ( x ) − g ( x ) ] d x . Calculation: The given functions are f ( x ) = x 3 − 6 x 2 + 9 x and y = x . The graph of the functions f ( x ) = x 3 − 6 x 2 + 9 x and y = x is shown below in Figure 1. From Figure 1, it is observed that the functions f ( x ) = x 3 − 6 x 2 + 9 x and y = x intersect at the points ( 0 , 0 ) , ( 2 , 2 ) and ( 4 , 4 ) . The function f ( x ) = x 3 − 6 x 2 + 9 x is greater than y = x in the interval [ 0 , 2 ] and the function f ( x ) = x 3 − 6 x 2 + 9 x is lesser than y = x in the interval [ 2 , 4 ] . By the above mentioned theorem, the area of the shaded region is computed as follows (B) To determine To find: The area bounded by the graphs of the equation f ( x ) = x 3 − 6 x 2 + 9 x and y = x + 1 .

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Author: Pearson

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Chapter 6, Problem 30RE

(A)

To determine

To find: The area bounded by the graphs of the equation f(x)=x3−6x2+9x and y=x.

(B)

To determine

To find: The area bounded by the graphs of the equation f(x)=x3−6x2+9x and y=x+1.

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Chapter 6, Problem 29RE

Chapter 6, Problem 31RE

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The area bounded by the graphs of the equation f ( x ) = x 3 − 6 x 2 + 9 x and y = x .

Solution Summary: The area bounded by the graphs of the equation f(x) is 8 square unit.