Skip to main content
Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.6.9
Chapter 6, Problem 6.6.9

Find the area of the surface generated when the given curve is revolved about the given axis.


y=8√x, for 9≤x≤20; about the x-axis

Verified step by step guidance
1
Step 1: Recall the formula for the surface area of a curve revolved about the x-axis. The formula is: S = 2π ∫ab y √(1 + (dy/dx)2) dx, where y is the function being revolved, and dy/dx is its derivative.
Step 2: Identify the given function y = 8√x and compute its derivative dy/dx. Using the power rule, dy/dx = 8 * (1/2) * x-1/2 = 4x-1/2.
Step 3: Substitute y = 8√x and dy/dx = 4x-1/2 into the surface area formula. This gives: S = 2π ∫920 (8√x) √(1 + (4x-1/2)2) dx.
Step 4: Simplify the expression inside the square root. Compute (4x-1/2)2 = 16x-1, so the integrand becomes: S = 2π ∫920 (8√x) √(1 + 16x-1) dx.
Step 5: Set up the integral for evaluation. The integral is now ready to be computed either analytically or numerically: S = 2π ∫920 (8√x) √(1 + 16/x) dx. Break this into manageable parts or use numerical methods to evaluate the integral.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Surface Area of Revolution

The surface area of revolution is calculated by rotating a curve around an axis. The formula involves integrating the length of the curve multiplied by the radius of rotation. For a curve y=f(x) revolved around the x-axis, the surface area S can be expressed as S = 2π ∫[a to b] f(x) √(1 + (f'(x))^2) dx, where f'(x) is the derivative of f(x).
Recommended video:
09:07
Example 1: Minimizing Surface Area

Parametric Representation

In some cases, curves can be represented parametrically, which involves expressing the coordinates of points on the curve as functions of a parameter. For the curve y=8√x, it can be useful to express x in terms of a parameter t, where t=x, to facilitate integration. This representation helps in calculating derivatives and applying the surface area formula effectively.
Recommended video:
Guided course
08:02
Parameterizing Equations

Definite Integral

A definite integral calculates the accumulation of quantities, such as area, over a specific interval. In this context, the limits of integration (9 to 20) define the section of the curve being revolved. Understanding how to evaluate definite integrals is crucial for finding the total surface area generated by the revolution of the curve around the x-axis.
Recommended video:
05:43
Definition of the Definite Integral
Related Practice
Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x³,y=0, and x=2; about the x-axis

1
views
Textbook Question

Why is integration used to find the work required to pump water out of a tank?

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

1
views
Textbook Question

Find the area of the region described in the following exercises.


The region in the first quadrant bounded by y=5/2−1/x and y=x

1
views
Textbook Question

Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.