The diagram shows the graph $y=e^{x}$ for $0 \leq x \leq 4$ - HSC - SSCE Mathematics Extension 2 - Question 5 - 2018 - Paper 1
Question 5
The diagram shows the graph $y=e^{x}$ for $0 \leq x \leq 4$. The region bounded by $y=-1$, $y=e^{x}$, $x=0$ and $x=4$ is rotated about the line $y=-1$ to form a soli... show full transcript
Worked Solution & Example Answer:The diagram shows the graph $y=e^{x}$ for $0 \leq x \leq 4$ - HSC - SSCE Mathematics Extension 2 - Question 5 - 2018 - Paper 1
Step 1
Identify the boundaries of the solid
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Answer
The solid is formed by rotating the area between the curve y=ex, the line y=−1, and the vertical lines x=0 and x=4. This defines a solid with a radius that depends on the distance from the line y=−1 to the curve.
Step 2
Determine the radius of the solid
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Answer
The radius of a cross-section at point x is given by the distance from y=−1 to y=ex. Therefore, the radius can be expressed as:
R(x)=ex−(−1)=ex+1
Step 3
Set up the volume integral using the disk method
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Answer
The volume V of the solid obtained by rotating the area around the line y=−1 can be calculated using the disk method, leading to the integral:
V=π∫04(R(x))2dx=π∫04(ex+1)2dx
Step 4
Conclusion
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Answer
Hence, the correct integral representing the volume of the solid formed is:
