Surface Area

Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.

x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

검증된 단계별 안내

Identify the parametric equations given: \(x = t + \sqrt{2}\) and \(y = \frac{t^{2}}{2} + \sqrt{2}t\), with the parameter \(t\) ranging from \(-\sqrt{2}\) to \(\sqrt{2}\).

Since the curve is revolved about the \(y\)-axis, use the formula for the surface area of a parametric curve revolved around the \(y\)-axis: \[ S = \int_{a}^{b} 2\pi |x(t)| \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]

Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\): - \(\frac{dx}{dt} = 1\) - \(\frac{dy}{dt} = t + \sqrt{2}\)

Substitute \(x(t)\), \(\frac{dx}{dt}\), and \(\frac{dy}{dt}\) into the surface area integral: \[ S = \int_{-\sqrt{2}}^{\sqrt{2}} 2\pi |t + \sqrt{2}| \sqrt{1^2 + (t + \sqrt{2})^2} \, dt \]

Evaluate the integral over the interval \([-\sqrt{2}, \sqrt{2}]\) to find the total surface area. Consider the absolute value in \(|t + \sqrt{2}|\) when setting up the integral, possibly splitting the integral if needed.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Understanding how to work with x(t) and y(t) allows us to describe complex curves and is essential for setting up integrals in problems involving curves defined parametrically.

Surface Area of Revolution

The surface area generated by revolving a curve around an axis is found using an integral formula that involves the radius of revolution and the arc length element. For parametric curves, the formula integrates 2π times the radius times the differential arc length over the parameter interval.

Arc Length of Parametric Curves

Arc length for parametric curves is computed by integrating the square root of the sum of the squares of the derivatives of x and y with respect to the parameter. This measure is crucial for calculating surface areas and other quantities involving the length of the curve.

Surface AreaFind the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis