Centroid:
Find the centroid of the region cut from the first quadrant by the curve
y = 1/√(x + 1) and the line x = 3.

검증된 단계별 안내

Identify the region bounded by the curve \(y = \frac{1}{\sqrt{x + 1}}\), the vertical line \(x = 3\), and the coordinate axes in the first quadrant. The region lies between \(x = 0\) and \(x = 3\), and above the \(x\)-axis.

Set up the integral for the area \(A\) of the region using the formula \(A = \int_0^3 y \, dx = \int_0^3 \frac{1}{\sqrt{x + 1}} \, dx\). This will give the total area under the curve from \(x=0\) to \(x=3\).

Find the coordinates of the centroid \((\bar{x}, \bar{y})\) using the formulas: \(\bar{x} = \frac{1}{A} \int_0^3 x y \, dx = \frac{1}{A} \int_0^3 x \frac{1}{\sqrt{x + 1}} \, dx\) and \(\bar{y} = \frac{1}{2A} \int_0^3 y^2 \, dx = \frac{1}{2A} \int_0^3 \left(\frac{1}{\sqrt{x + 1}}\right)^2 \, dx\).

Evaluate each integral separately: the area integral, the \(x\)-moment integral \(\int_0^3 x y \, dx\), and the \(y\)-moment integral \(\int_0^3 y^2 \, dx\). Use substitution if necessary, for example, let \(u = x + 1\) to simplify the integrals.

After computing the integrals, substitute the results back into the centroid formulas to find \(\bar{x}\) and \(\bar{y}\). These values give the coordinates of the centroid of the region.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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

Centroid of a Region

The centroid is the geometric center or 'balance point' of a plane region. It is found by calculating the average position of all points in the area, typically using integrals to find the coordinates (x̄, ȳ). For regions bounded by curves, the centroid coordinates are given by the moments divided by the area.

Definite Integrals for Area and Moments

Definite integrals are used to compute the area under a curve and the moments about the axes. The area is found by integrating the function over the given interval, while moments involve integrating the product of the function and x or y coordinates. These integrals are essential to determine the centroid coordinates.

Region Bounded by a Curve and a Vertical Line

The region is defined by the curve y = 1/√(x + 1), the vertical line x = 3, and the coordinate axes in the first quadrant. Understanding the limits of integration and the shape of the region is crucial for setting up the correct integrals to find area and moments.

Centroid:Find the centroid of the region cut from the first quadrant by the curvey = 1/√(x + 1) and the line x = 3.