41–48. Geometry problems Use a table of integrals to solve the following problems.
43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.

검증된 단계별 안내

Step 1: Recall the formula for the arc length of a curve y = f(x) on the interval [a, b]. The formula is given by:

\int_{a}^{b} \sqrt{1 + {\frac{d y}{d x}}^{2}} d x

Step 2: Compute the derivative of y = eˣ with respect to x. The derivative is

\frac{d y}{d x}

= eˣ.

Step 3: Substitute the derivative into the arc length formula. The integrand becomes

\sqrt{1 + e^{2}}

Step 4: Set up the definite integral for the arc length on the interval [0, ln 2]. The integral is:

\int_{0}^{ln 2} \sqrt{1 + e^{2}} d x

Step 5: Use a table of integrals to evaluate the integral. Look for an entry that matches the form of the integrand

\sqrt{1 + e^{2}}

, and apply the corresponding formula to find the arc length.

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주요 개념

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Arc Length Formula

The arc length formula is used to calculate the length of a curve defined by a function y = f(x) over a specific interval [a, b]. It is given by the integral L = ∫ from a to b √(1 + (dy/dx)²) dx, where dy/dx is the derivative of the function. This formula accounts for the changes in both x and y as you move along the curve.

Integration Techniques

Integration techniques are methods used to evaluate integrals, which are essential for finding areas under curves or lengths of curves. Common techniques include substitution, integration by parts, and using tables of integrals. In this problem, using a table of integrals can simplify the process of finding the integral needed to calculate the arc length.

Exponential Functions

Exponential functions, such as y = eˣ, are functions where the variable appears in the exponent. They have unique properties, including a constant rate of growth and a derivative that is equal to the function itself. Understanding the behavior of exponential functions is crucial for calculating their derivatives and applying them in the arc length formula.

41–48. Geometry problems Use a table of integrals to solve the following problems.43. Find the length of the curve y = eˣ on the interval from 0 to ln 2.