20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.

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Identify the function and the interval: The curve is given by \(y = \frac{1}{2\sqrt{x}}\) and the interval is from \(x = \frac{1}{4}\) to \(x = 4\).

Recall the formula for the volume of a solid of revolution about the x-axis: \(V = \pi \int_{a}^{b} [f(x)]^{2} \, dx\), where \(f(x)\) is the function being revolved.

Square the function inside the integral: \(\left( \frac{1}{2\sqrt{x}} \right)^{2} = \frac{1}{4x}\).

Set up the integral for the volume: \(V = \pi \int_{\frac{1}{4}}^{4} \frac{1}{4x} \, dx\).

Evaluate the integral by integrating \(\frac{1}{4x}\) with respect to \(x\) over the interval \(\left[ \frac{1}{4}, 4 \right]\), then multiply the result by \(\pi\) to find the volume.

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주요 개념

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Solid of Revolution

A solid of revolution is formed when a plane region is rotated about a line (axis), creating a three-dimensional object. The volume of such solids can be found using integral calculus by summing infinitesimal disks or washers perpendicular to the axis of rotation.

Disk Method

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

Definite Integration

Definite integration computes the exact accumulation of quantities, such as area or volume, over a specific interval. In this problem, integrating the squared function from x=1/4 to x=4 calculates the total volume generated by revolving the region around the x-axis.

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.a. Find the volume of the solid.