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

Previous problemNext problem

1:58 minutes

Problem 6.6.23a

Textbook Question

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

a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.

Verified step by step guidance

Step 1: Understand the problem context. The problem involves finding the surface area generated by revolving a curve about the y-axis. The curve is given by y = f(x) on the interval [a, b]. We need to verify if the given integral expression for the surface area is correct.

Step 2: Recall the formula for the surface area when revolving a curve about the y-axis. If the curve is expressed as x = g(y) (i.e., x as a function of y), then the surface area S is given by: $S = \int_{c}^{d} 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy$ where [c, d] is the interval for y.

Step 3: Note that the problem gives the curve as y = f(x), so to use the formula revolving around the y-axis in terms of y, we need to express x as a function of y, i.e., find the inverse function x = f^{-1}(y). Then, the derivative $\frac{dx}{dy}$ is the derivative of the inverse function.

Step 4: Compare the given integral $\int_{f(b)}^{f(a)} 2\pi f(y) \sqrt{1 + (f'(y))^2} \, dy$ with the correct formula. Notice that the integrand uses $f(y)$ and $f'(y)$, but since f is originally a function of x, $f(y)$ and $f'(y)$ do not make sense unless f is inverted. This suggests the given integral is not correctly formulated.

Step 5: Conclusion: The given integral is not the correct formula for the surface area generated by revolving y = f(x) about the y-axis. The correct approach requires expressing x as a function of y and using the formula involving $x$ and $\frac{dx}{dy}$. Therefore, the statement is false.

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Key Concepts

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

Surface Area of Revolution

The surface area generated by revolving a curve around an axis is found by integrating the circumference of circular slices times the arc length element. For revolution about the y-axis, the formula involves integrating 2π times the radius (x or function of y) multiplied by the differential arc length along y.

Parametrization and Variable of Integration

When revolving a curve about the y-axis, the integral is typically expressed in terms of y, requiring the function to be rewritten as x = g(y). The limits of integration correspond to y-values, and the derivative inside the integral must be with respect to y, not x.

Arc Length Element in Terms of y

The differential arc length element ds is given by √(1 + (dx/dy)^2) dy when integrating with respect to y. Using f'(y) instead of dx/dy or mixing variables leads to incorrect formulas, so careful differentiation and substitution are essential.

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Textbook Question

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

b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.

Textbook Question

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

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

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

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