Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.

b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?

검증된 단계별 안내

Step 1: Understand the problem. The region R is bounded by the graphs of y = 1 + √x, x = 4, and y = 1. When this region is revolved about the x-axis, the resulting solid has cross sections that are washers. The goal is to determine the inner radius of a washer at a point x in the interval [0, 4].

Step 2: Recall the formula for the inner radius of a washer. The inner radius is the distance from the x-axis to the inner curve of the region being revolved. In this case, the inner curve is the horizontal line y = 1.

Step 3: Analyze the geometry of the region. At any point x in [0, 4], the inner radius is determined by the vertical distance from the x-axis to the line y = 1. Since the line y = 1 is constant, the inner radius does not depend on x.

Step 4: Express the inner radius mathematically. The inner radius is simply the value of y for the inner curve, which is y = 1. Therefore, the inner radius is 1 for all x in the interval [0, 4].

Step 5: Conclude that the inner radius of the washer at any point x in [0, 4] is constant and equal to 1. This is because the inner curve is a horizontal line at y = 1.

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주요 개념

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

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this case, the region R is revolved around the x-axis, forming a solid. Understanding this concept is crucial for visualizing the shape and determining its volume using integration techniques.

Washer Method

The washer method is a technique used to find the volume of a solid of revolution when the cross-sections are washers. It involves subtracting the volume of the inner solid from the outer solid. The inner radius and outer radius are essential for setting up the integral that calculates the volume of the solid formed by revolving the region around the axis.

Inner Radius

The inner radius in the context of the washer method refers to the distance from the axis of rotation to the inner curve of the region being revolved. For the given problem, it is the distance from the x-axis to the curve defined by y=1. Identifying this radius is necessary to accurately compute the volume of the solid formed by the revolution.

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises. Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers. b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?