Question

Determine whether the following statements are true and give an explanation or counterexample.

a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.

Accepted Answer

Step 1: Understand the problem context. The problem involves finding the surface area generated by revolving a curve about the y-axis. The curve is given by y = f(x) on the interval [a, b]. We need to verify if the given integral expression for the surface area is correct. Step 2: Recall the formula for the surface area when revolving a curve about the y-axis. If the curve is expressed as x = g(y) (i.e., x as a function of y), then the surface area S is given by:

$$S = \int_{c}^{d} 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\$$right)^2$$} \, dy$$

where [c, d] is the interval for y. Step 3: Note that the problem gives the curve as y = f(x), so to use the formula revolving around the y-axis in terms of y, we need to express x as a function of y, i.e., find the inverse function x = $$f^{-1}(y). $$Then, the derivative $$\frac{dx}{dy} is $$the derivative of the inverse function. Step 4: Compare the given integral $$\int_{f(b)}^{f(a)} 2\pi f(y) \sqrt{1 + (f'(y))^2$} \, dy$$ with the correct formula. Notice that the integrand uses $$f(y)$$ and $$f'(y)$$, but since f is originally a function of x, $$f(y)$$ and $$f'(y) do $$not make sense unless f is inverted. This suggests the given integral is not correctly formulated. Step 5: Conclusion: The given integral is not the correct formula for the surface area generated by revolving y = f(x) about the y-axis. The correct approach requires expressing x as a function of y and using the formula involving $$x$$ and $$\frac{dx}{dy}. $$Therefore, the statement is false.

Determine whether the following statements are true and give an explanation or counterexample.

a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.

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Determine whether the following statements are true and give an explanation or counterexample. a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.

Verified video answer for a similar problem:

Key Concepts

Surface Area of Revolution

Parametrization and Variable of Integration

Arc Length Element in Terms of y

Determine whether the following statements are true and give an explanation or counterexample.

a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.