Problem

The function
f(x)=x4x2+8x
is negative on (3,4) and positive on (4,5). Find the area of the region bounded by f(x), the x-axis, and the vertical lines x=3 and x=5. Round to 2 decimal places.
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Final Answer: The area of the region bounded by the function f(x), the x-axis, and the vertical lines x=3 and x=5 is 0.02.

Steps

Step 1 :Define the function f(x)=x4x2+8x.

Step 2 :Calculate the integral of the function from 3 to 4. The absolute value is taken because the function is negative in this interval.

Step 3 :The integral from 3 to 4 is approximately 0.013323970741445814.

Step 4 :Calculate the integral of the function from 4 to 5. The function is positive in this interval, so the absolute value is not necessary.

Step 5 :The integral from 4 to 5 is approximately 0.008492285853199761.

Step 6 :Add the two areas to get the total area of the region bounded by the function, the x-axis, and the vertical lines x=3 and x=5.

Step 7 :The total area is approximately 0.02.

Step 8 :Final Answer: The area of the region bounded by the function f(x), the x-axis, and the vertical lines x=3 and x=5 is 0.02.

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