39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

x =2

검증된 단계별 안내

Identify the region R bounded by the curves: \(y = x^{2}\), \(x = 1\), and \(y = 0\). This region lies between \(x=0\) and \(x=1\) above the \(x\)-axis.

Since we are revolving the region about the vertical line \(x = 2\), use the shell method with vertical shells parallel to the axis of revolution. The height of each shell is given by the function \(y = x^{2}\).

Determine the radius of a typical shell. The radius is the horizontal distance from the shell at position \(x\) to the line \(x=2\), which is \(2 - x\).

Write the volume element of a shell as \(dV = 2\pi \times (\text{radius}) \times (\text{height}) \times (\text{thickness}) = 2\pi (2 - x)(x^{2}) \, dx\).

Set up the integral for the volume by integrating \(dV\) from \(x=0\) to \(x=1\): \(V = \int_{0}^{1} 2\pi (2 - x)(x^{2}) \, dx\). This integral represents the volume of the solid generated.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is found by multiplying its circumference, height, and thickness. This method is especially useful when the axis of rotation is parallel to the axis of the function being integrated.

Setting up the Shell Radius and Height

When revolving around a vertical line like x=2, the radius of each shell is the horizontal distance from the shell to the line x=2. The height of the shell corresponds to the function value, here y = x², bounded by y=0. Correctly identifying radius and height expressions is crucial for the integral setup.

Limits of Integration

The limits of integration correspond to the interval over which the region extends along the axis perpendicular to the shells. Since the region is bounded by x=0 (implied by y=0 and x²) and x=1, the integral limits for x run from 0 to 1. Proper limits ensure the volume calculation covers the entire region.

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.x =2