Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).

b. Find the volume of the solid generated when R is revolved about the x-axis.

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Identify the function and the region: The function is given as \(f(x) = (9 + x^{2})^{-1/2}\), and the region \(R\) is bounded by this curve, the x-axis, and the vertical lines \(x=0\) and \(x=4\).

Recall the formula for the volume of a solid of revolution about the x-axis: When a region bounded by \(y=f(x)\) is revolved about the x-axis, the volume \(V\) is given by the integral \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\).

Set up the integral for the volume: Substitute \(f(x)\) into the formula, so the volume is \(V = \pi \int_{0}^{4} \left((9 + x^{2})^{-1/2}\right)^{2} \, dx\).

Simplify the integrand: Since squaring \((9 + x^{2})^{-1/2}\) gives \((9 + x^{2})^{-1}\), the integral becomes \(V = \pi \int_{0}^{4} \frac{1}{9 + x^{2}} \, dx\).

Evaluate the integral: Recognize that the integral of \(\frac{1}{a^{2} + x^{2}}\) with respect to \(x\) is \(\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\). Use this to express the definite integral from 0 to 4, then multiply by \(\pi\) to find the volume.

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주요 개념

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Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. The volume can be computed using methods like the disk/washer or shell method, which integrate cross-sectional areas perpendicular to the axis of rotation.

Disk Method

The disk method calculates volume by slicing the solid perpendicular to the axis of rotation into thin disks. Each disk's volume is approximated by π(radius)^2 times thickness, and integrating these volumes over the interval gives the total volume.

Integration of Functions with Radical Expressions

Integrating functions like f(x) = (9 + x²)^(-1/2) requires understanding how to handle radicals and powers in integrals. Recognizing this as a form related to inverse trigonometric functions or using substitution simplifies the integration process.

Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).b. Find the volume of the solid generated when R is revolved about the x-axis.