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How to find area of this curved shape?

curved shape

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It looks like the left and right boundaries of your region can be represented by equations of the form $x = f(y)$ and $x = g(y)$ for $0 \le y \le 16$. Then your area is $\int_0^{16} (g(y) - f(y))\ dy$. It also looks like $g(y) - f(y)$ is approximately (but not exactly) $4 + X$, so the area is approximately $16 (4 + X)$.

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If the claim is that the cross-sections are all $4+x$ wide, then by Cavalieri's principle the area is $16(4+x)$.

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