Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=x and y=4√x; about the x-axis

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Identify the region R bounded by the curves \(y = x\) and \(y = 4\sqrt{x}\), and the axis of revolution, which is the x-axis.

Find the points of intersection of the curves by setting \(x = 4\sqrt{x}\). Solve for \(x\) to determine the limits of integration.

Since the region is revolved around the x-axis, use the washer method to find the volume. The volume element is given by \(\pi \int_a^b \left(R(x)^2 - r(x)^2\right) \, dx\), where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius.

Determine which curve is the outer radius and which is the inner radius relative to the x-axis. Here, the outer radius is the curve farther from the x-axis, and the inner radius is the closer curve. Express these radii as functions of \(x\).

Set up the integral for the volume using the limits of integration found in step 2 and the radii from step 4. The integral will be \(V = \pi \int_a^b \left( (4\sqrt{x})^2 - (x)^2 \right) \, dx\). Evaluate this integral to find the volume.

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주요 개념

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Finding the Region of Integration

To find the volume of the solid, first identify the region bounded by the curves y = x and y = 4√x. Determine their points of intersection by solving x = 4√x, which sets the limits of integration along the x-axis. This step is crucial to correctly define the interval over which the volume will be calculated.

Volume of Solids of Revolution Using the Washer Method

When a region is revolved around the x-axis, the volume can be found using the washer method. This involves integrating the difference of the squares of the outer and inner radii (functions of x) multiplied by π over the interval. Here, the outer radius is the upper curve and the inner radius is the lower curve relative to the axis of rotation.

Setting up and Evaluating the Definite Integral

After determining the limits and radii, set up the definite integral for volume: V = π∫[a to b] (R(x))^2 - (r(x))^2 dx. Carefully substitute the expressions for R(x) and r(x) from the given curves, then evaluate the integral using appropriate techniques such as substitution or power rule to find the exact volume.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.y=x and y=4√x; about the x-axis