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Numerade Educator

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Problem 33 Medium Difficulty

If the curve $ y = f(x) $, $ a \le x \le b $, is rotated about the horizontal line $ y = c $, where $ f(x) \le c $, find a formula for the area of the resulting surface.

Answer

$$\int_{a}^{b} 2 \pi(c-f(x)) \sqrt{1+\left\{\frac{d y}{d x}\right\}^{2}} d x$$

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Video Transcript

this question asked us to find a formula for the area of the resulting surface. What we know we're gonna be looking at is the fact that we have two times the distance from the line times, the ark function times, DEA ex. Obviously, we always under into girls with DX. So our bounds we could just, you know what? These is a to B to pie. This is a given when we're writing the integral out, we know we have C minus half of acts. The reason why because we're looking at the distance from why equals C of any point X comma after Vex, you can consider f of X to essentially be or why, But in this context we're using a fullback because we're writing this in terms of X. Then we know this is gonna be multiplied by one plus driven of, and the derivative under the square root is squared. And as I said, you always end this India vax