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If the infinite curve \( y=e^{-7 x}, x \geq 0 \), is rotated about the \( x \)-axis, find the area of the resulting surface. Need Help? Readit

Ask by George Hamilton. in the United States
Mar 15,2025

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The area of the surface formed by rotating the curve \( y = e^{-7x} \) for \( x \geq 0 \) about the \( x \)-axis is \( \frac{\pi}{49}\left[35\sqrt{2} + \ln\left(7+5\sqrt{2}\right)\right] \).

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The Deep Dive

To find the area of the surface generated by rotating the curve \( y = e^{-7x} \) about the \( x \)-axis, the formula for surface area is given by: \[ S = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] For \( y = e^{-7x} \): 1. Calculate \(\frac{dy}{dx} = -7e^{-7x}\). 2. Thus, \( \left(\frac{dy}{dx}\right)^2 = 49e^{-14x} \) resulted in: \[ 1 + \left(\frac{dy}{dx}\right)^2 = 1 + 49e^{-14x} \] 3. Therefore, the surface area becomes: \[ S = 2\pi \int_{0}^{\infty} e^{-7x} \sqrt{1 + 49e^{-14x}} \, dx \] 4. This integral can be solved using numerical methods or by substitution to find a more manageable form. Though it might look complicated, don't let it intimidate you! The beauty of calculus is in unraveling such intricacies layer by layer. As you compute, keep your favorite snack nearby for some extra motivation! 🍪

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