Find the volumes of the solids in Exercises 135 and 136.
135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are
b. vertical squares whose base edges run from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).

검증된 단계별 안내

Identify the interval over which the solid extends along the x-axis, which is from \(x = -1\) to \(x = 1\).

Determine the length of the base of each square cross-section at a given \(x\). The base runs vertically from \(y = -\frac{1}{\sqrt{1+x^2}}\) to \(y = \frac{1}{\sqrt{1+x^2}}\), so the base length is the difference between these two values.

Calculate the base length as \(\text{base} = \frac{1}{\sqrt{1+x^2}} - \left(-\frac{1}{\sqrt{1+x^2}}\right) = \frac{2}{\sqrt{1+x^2}}\).

Since the cross-sections are squares, the area of each cross-section is \(A(x) = \left(\text{base}\right)^2 = \left(\frac{2}{\sqrt{1+x^2}}\right)^2\).

Set up the volume integral by integrating the cross-sectional area along the x-axis: \(V = \int_{-1}^{1} A(x) \, dx = \int_{-1}^{1} \left(\frac{2}{\sqrt{1+x^2}}\right)^2 \, dx\). This integral will give the volume of the solid.

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.

Volume of a Solid with Known Cross-Sections

This method involves finding the volume of a solid by integrating the area of cross-sectional shapes perpendicular to an axis. For each x-value, the area of the cross-section is computed, then integrated over the given interval to find the total volume.

Determining the Side Length of the Square Cross-Section

The side length of each square cross-section is the vertical distance between the two curves y = -1/√(1+x²) and y = 1/√(1+x²). Calculating this length correctly is essential to find the area of the square at each x.

Definite Integration over the Interval

Once the area of the cross-section is expressed as a function of x, definite integration from x = -1 to x = 1 sums these areas to find the volume. Understanding how to set up and evaluate this integral is crucial.

Find the volumes of the solids in Exercises 135 and 136.135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis areb. vertical squares whose base edges run from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).