Length of a curve
Find the length of the curve
y = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.

검증된 단계별 안내

Recognize that the curve is defined by an integral function: \(y = \int_{1}^{x} \sqrt{\sqrt{t} - 1} \, dt\). This means \(y\) is given as a function of \(x\) through an integral with variable upper limit.

Recall the formula for the length \(L\) of a curve \(y = f(x)\) from \(x = a\) to \(x = b\): \(L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\).

Use the Fundamental Theorem of Calculus to find \(\frac{dy}{dx}\). Since \(y = \int_{1}^{x} g(t) \, dt\) where \(g(t) = \sqrt{\sqrt{t} - 1}\), then \(\frac{dy}{dx} = g(x) = \sqrt{\sqrt{x} - 1}\).

Substitute \(\frac{dy}{dx}\) into the length formula: \(L = \int_{1}^{16} \sqrt{1 + \left(\sqrt{\sqrt{x} - 1}\right)^2} \, dx\).

Simplify the expression inside the square root and set up the integral for evaluation: \(L = \int_{1}^{16} \sqrt{1 + (\sqrt{x} - 1)} \, dx = \int_{1}^{16} \sqrt{\sqrt{x}} \, dx\). This integral can then be evaluated using substitution or power rules.

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.

Definition of the Length of a Curve

The length of a curve y = f(x) from x = a to x = b is given by the integral of the square root of 1 plus the square of the derivative, ∫_a^b √(1 + (dy/dx)^2) dx. This formula measures the distance along the curve between two points.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if y = ∫_a^x g(t) dt, then dy/dx = g(x). This allows us to find the derivative of an integral-defined function, which is essential for computing dy/dx in the curve length formula.

Chain Rule and Simplification of Derivatives

When differentiating functions involving nested expressions, the chain rule helps find the derivative accurately. In this problem, simplifying dy/dx = sqrt(sqrt(x) - 1) is crucial before substituting into the length formula to evaluate the integral efficiently.

Length of a curveFind the length of the curvey = ∫(from 1 to x) sqrt(sqrt(t) - 1) dt, where 1 ≤ x ≤ 16.