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.

d. Write an integral for the volume of the solid.

검증된 단계별 안내

Step 1: Identify the boundaries of the region R. The region is bounded by the curves y = 1 + √x, x = 4, and y = 1. The curve y = 1 + √x is the upper boundary, and y = 1 is the lower boundary.

Step 2: Recognize that the solid is formed by revolving the region R about the x-axis. The cross-sections of the solid are washers, which means the volume can be calculated using the washer method.

Step 3: Write the formula for the volume using the washer method. The volume is given by the integral: V = ∫[a to b] π[(outer radius)^2 - (inner radius)^2] dx. Here, the outer radius is the distance from the x-axis to the curve y = 1 + √x, and the inner radius is the distance from the x-axis to the line y = 1.

Step 4: Determine the limits of integration. The region R spans from x = 0 to x = 4, so the limits of integration are a = 0 and b = 4.

Step 5: Substitute the expressions for the outer and inner radii into the formula. The outer radius is y = 1 + √x, and the inner radius is y = 1. The integral becomes: V = ∫[0 to 4] π[(1 + √x)^2 - (1)^2] dx.

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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 how to calculate its volume.

Washer Method

The washer method is a technique used to find the volume of a solid of revolution when the cross-sections perpendicular to the axis of rotation are washers (disks with holes). This method involves integrating the area of the washers, which is calculated as the difference between the outer and inner radii squared, multiplied by π, over the interval of integration.

Definite Integral

A definite integral represents the accumulation of quantities, such as area or volume, over a specific interval. In this context, it is used to calculate the volume of the solid formed by revolving region R around the x-axis. The limits of integration correspond to the bounds of the region, and the integrand is derived from the washer method.

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. d. Write an integral for the volume of the solid.