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

Introduction to Volume & Disk Method

Calculus

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

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1:58 minutes

Problem 6.5.19

Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

Verified step by step guidance

Rewrite the given curve equation in terms of $ y $: $ x = 2y - 4 $. Since $ x $ is expressed as a function of $ y $, we will use the arc length formula with respect to $ y $.

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

Calculate the derivative $ \frac{dx}{dy} $ from the given function: $$ \frac{dx}{dy} = \frac{d}{dy}(2y - 4) = 2 $$

Substitute $ \frac{dx}{dy} = 2 $ into the arc length integral and set the limits from $ y = -3 $ to $ y = 4 $: $$ L = \int_{-3}^{4} \sqrt{1 + (2)^2} \, dy = \int_{-3}^{4} \sqrt{1 + 4} \, dy = \int_{-3}^{4} \sqrt{5} \, dy $$

Evaluate the integral by factoring out the constant $ \sqrt{5} $: $$ L = \sqrt{5} \int_{-3}^{4} dy = \sqrt{5} [y]_{-3}^{4} = \sqrt{5} (4 - (-3)) = \sqrt{5} \times 7 $$ This gives the arc length in terms of $ \sqrt{5} $ and the interval 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 of a curve defined by a function can be found using the integral formula: L = ∫√(1 + (dy/dx)²) dx or L = ∫√(1 + (dx/dy)²) dy, depending on the variable of integration. This formula sums infinitesimal line segments along the curve to find its total length.

Parametric and Implicit Differentiation

When the curve is given in terms of y or implicitly, it is important to correctly find the derivative dx/dy or dy/dx. This may involve rearranging the equation or using implicit differentiation to express the derivative needed for the arc length formula.

Geometric Interpretation and Verification

After calculating the arc length using calculus, it is useful to verify the result geometrically if possible. For linear or simple curves, the arc length corresponds to the distance between endpoints, providing a way to check the correctness of the integral calculation.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.

Textbook Question

Consider the following curves on the given intervals.

b. Use a calculator or software to approximate the surface area.

y=cos x, for 0≤x≤π/2; about the x-axis

Textbook Question

Consider the following curves on the given intervals.

a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis.

y=tan x , for 0≤x≤π/4; about the x-axis

Textbook Question

Consider the following curves on the given intervals.

b. Use a calculator or software to approximate the surface area.

y=tan x , for 0≤x≤π/4; about the x-axis

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Textbook Question

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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y = x^3/2 / 3 − x^1/2 on [4, 16]

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9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = (x²+2)^3/2 / 3 on [0, 1]

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9–20. Arc length calculations Find the arc length of the following curves on the given interval.x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

Key Concepts

Arc Length Formula

Parametric and Implicit Differentiation

Geometric Interpretation and Verification

Watch next

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)