Revolve the region bounded by $y = e ^{ x }$, the $x$-axis, and the lines $x=0$ and $x=1$ about the line $y=-3$. Find the volume.
$V=[?]$
Round your answer to the nearest thousandth.
Answer (2)
Determine the outer radius R ( x ) = e x + 3 and inner radius r ( x ) = 3 when revolving the region around y = − 3 .
Set up the volume integral using the washer method: V = π ∫ 0 1 (( e x + 3 ) 2 − 3 2 ) d x .
Evaluate the integral: V = π [ 2 1 e 2 x + 6 e x ] 0 1 = π ( 2 1 e 2 + 6 e − 2 13 ) .
Approximate the volume to the nearest thousandth: 42.425 .
Explanation
Problem Setup We are asked to find the volume of the solid generated by revolving the region bounded by y = e x , the x -axis, and the lines x = 0 and x = 1 about the line y = − 3 . We will use the washer method to find the volume.
Finding Radii The outer radius R ( x ) is the distance from the axis of revolution y = − 3 to the curve y = e x , so R ( x ) = e x − ( − 3 ) = e x + 3 . The inner radius r ( x ) is the distance from the axis of revolution y = − 3 to the x -axis ( y = 0 ), so r ( x ) = 0 − ( − 3 ) = 3 .
Setting up the Integral The volume is given by the integral V = π ∫ 0 1 ( R ( x ) 2 − r ( x ) 2 ) d x = π ∫ 0 1 (( e x + 3 ) 2 − 3 2 ) d x . Expanding the integrand, we get ( e x + 3 ) 2 − 3 2 = e 2 x + 6 e x + 9 − 9 = e 2 x + 6 e x .
Evaluating the Integral Now we evaluate the integral: V = π ∫ 0 1 ( e 2 x + 6 e x ) d x = π [ 2 1 e 2 x + 6 e x ] 0 1 . Evaluating the limits of integration, we get V = π [( 2 1 e 2 ( 1 ) + 6 e 1 ) − ( 2 1 e 2 ( 0 ) + 6 e 0 )] = π [( 2 1 e 2 + 6 e ) − ( 2 1 + 6 )] = π ( 2 1 e 2 + 6 e − 2 13 ) .
Final Calculation Approximating the volume, we have V ≈ π ( 2 1 ( 7.389 ) + 6 ( 2.718 ) − 6.5 ) ≈ π ( 3.6945 + 16.308 − 6.5 ) ≈ π ( 13.5025 ) ≈ 42.42475526638944 . Rounding to the nearest thousandth, we get V ≈ 42.425 .
Final Answer The volume of the solid generated by revolving the region is approximately 42.425 .
Examples
Imagine you are designing a custom-shaped fuel tank for a rocket. The tank's shape is formed by revolving a curve around an axis. Calculating the volume of this tank, as we did in this problem, is crucial to know how much fuel it can hold. This ensures the rocket has enough fuel for its mission, highlighting the practical importance of volume calculations in engineering.
The volume of the solid formed by revolving the region bounded by y = e x , the x -axis, and the lines x = 0 to x = 1 about y = − 3 is calculated using the washer method and found to be approximately 42.425 .
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