The area bounded by 𝑥=𝑦2−2 and 𝑥=𝑒𝑦 between 𝑦=−1 and 𝑦=1 is revolved about the line 𝑦=1. The Volume integral for the solid generated is
Question
The area bounded by x = y^2 - 2 and x = e^y between y = -1 and y = 1 is revolved about the line y = 1. The Volume integral for the solid generated is
Solution
The volume of a solid of revolution can be found using the formula for the volume of a solid of revolution about the y-axis:
V = ∫[a,b] π (outer function)^2 - π (inner function)^2 dy
Here, the outer function is ey and the inner function is y^2 - 2. The limits of integration are y = -1 to y = 1.
So, the volume integral for the solid generated is:
V = ∫[-1,1] π (e^y)^2 - π (y^2 - 2)^2 dy
This integral can be simplified to:
V = ∫[-1,1] π (e^2y - (y^4 - 4y^2 + 4)) dy
This integral can be
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