Question

120. Equal volumesa. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?b. Repeat part (a) on the interval [0, 1].

Accepted Answer

Identify the region R bounded by the curve $$f(x) = x$$^{-p}$$ and the x-axis for x$$ \geq 1$$ (part a) or x$$ \in [0,1]$$ (part b). This means the region lies between the curve and the x-axis over the specified interval. Express the volume V_1$ generated by revolving R about the x-axis using the disk/washer method. The formula is:

V_1$$ = \pi \int_a^b [f(x)]^2 \, dx = \pi \int_a^b x^{-2p} \, dx,$$

where $$[a,b]$$ is the interval of integration (either $$[1, \infty)$$ for part a or $$[0,1]$$ for part b). Express the volume V_2$ generated by revolving R about the y-axis using the shell method. The formula is:

V_2 = 2$$\pi \int_a^b x \cdot f(x) \, dx = 2\pi \int_a^b x \cdot x^{-p} \, dx = 2\pi \int_a^b x^{1-p} \, dx,$$

with the same interval $$[a,b]$$ as above. Determine the conditions on p$ for which both V_1$ and V_2$ converge (i.e., the integrals are finite). This involves analyzing the improper integrals if the interval is infinite or near zero, and using the convergence criteria for integrals of the form $$\int x^m \, dx$$. Set the expressions for V_1$ and V_2$ equal to each other and solve for p$. This will involve evaluating the integrals symbolically (without computing the final numeric values), then equating the results and isolating p$ to find the values where V_1$ = V_2$.

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Key Concepts

Volumes of Solids of Revolution

Improper Integrals and Convergence

Function Behavior and Parameter Dependence