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 6.6.27b
Chapter 6, Problem 6.6.27b

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 

Verified step by step guidance
1
Step 1: Recall the formula for the surface area of a curve rotated about the x-axis. The formula is: S = 2π ∫[a,b] y √(1 + (dy/dx)^2) dx, where y is the function and dy/dx is its derivative.
Step 2: Identify the given function and interval. Here, y = tan(x) and the interval is [0, π/4]. Substitute y = tan(x) into the formula.
Step 3: Compute the derivative of y = tan(x). The derivative is dy/dx = sec^2(x). Substitute this into the formula for surface area.
Step 4: Simplify the integrand. The integrand becomes tan(x) √(1 + sec^4(x)). Set up the integral: S = 2π ∫[0,π/4] tan(x) √(1 + sec^4(x)) dx.
Step 5: Use a calculator or software to approximate the value of the integral numerically. This step involves evaluating the integral using numerical methods, as the integrand is complex and does not have a simple antiderivative.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
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 refers to the area of a surface created when a curve is rotated around an axis. For a function y = f(x) rotated about the x-axis, the formula involves integrating the circumference of infinitesimally thin circular slices of the surface. The formula is given by 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

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve. It is the reverse process of differentiation and is used to calculate quantities such as area, volume, and surface area. In the context of surface area, definite integrals are used to sum up the contributions of each infinitesimal slice of the surface.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Trigonometric Functions

Trigonometric functions, such as tangent, sine, and cosine, relate the angles of a triangle to the lengths of its sides. The function y = tan(x) specifically represents the ratio of the opposite side to the adjacent side in a right triangle. Understanding the behavior of these functions, especially within specific intervals, is crucial for accurately calculating areas and understanding the shape of the curves involved.
Recommended video:
6:04
Introduction to Trigonometric Functions
Related Practice
Textbook Question

55–58. Marginal cost Consider the following marginal cost functions.


b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.


C′(x)=200−0.05x

Textbook Question

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


b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

1
views
Textbook Question

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?

1
views
Textbook Question

40–43. Population growth


When records were first kept (t=0), the population of a rural town was 250 people. During the following years, the population grew at a rate of P′(t) = 30(1+√t), where t is measured in years.


b. Find the population P(t) at any time t≥0.

Textbook Question

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

b. The distance traveled between t=0 and t=5

Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 50e^−2t on [0, 4]