Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Area Between Curves

Calculus

9. Graphical Applications of Integrals

Area Between Curves

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2:26 minutes

Problem 8.7.43

Textbook Question

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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]

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(a) A(2)

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Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

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Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

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