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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 6.6.19
Chapter 6, Problem 6.6.19

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


x=√12y−y^2, for 2≤y≤10; about the y-axis

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Step 1: Recall the formula for the surface area of a curve revolved about the y-axis. The formula is: A = 2π ∫[a,b] x √(1 + (dx/dy)^2) dy, where x is expressed as a function of y, and the integral is evaluated over the given interval [a, b].
Step 2: Identify the given curve and interval. The curve is x = √(12y - y^2), and the interval is 2 ≤ y ≤ 10. Substitute this expression for x into the formula.
Step 3: Compute dx/dy. Differentiate x = √(12y - y^2) with respect to y using the chain rule. This gives dx/dy = (1/2)(12y - y^2)^(-1/2) * (12 - 2y). Simplify this derivative.
Step 4: Substitute dx/dy into the formula for surface area. The integrand becomes x √(1 + (dx/dy)^2), where x = √(12y - y^2) and dx/dy is the derivative computed in Step 3.
Step 5: Set up the definite integral for the surface area. The integral is A = 2π ∫[2,10] √(12y - y^2) √(1 + ((1/2)(12y - y^2)^(-1/2) * (12 - 2y))^2) dy. Simplify the integrand as much as possible, and then proceed to evaluate the integral using appropriate techniques such as substitution or numerical methods.

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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 of revolution is calculated by rotating a curve around an axis. The formula for the surface area generated by revolving a curve y = f(x) about the y-axis is given by S = 2π ∫ x * √(1 + (dy/dx)²) dy, where the integral is evaluated over the specified interval. This concept is essential for determining the area of the surface created by the rotation of the curve.
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Parametric Equations

In this problem, the curve is given in terms of y, which can be expressed as a function of y (x = √(12y - y²)). Understanding how to manipulate and interpret parametric equations is crucial for finding the derivatives needed in the surface area formula. This involves differentiating x with respect to y to find dy/dx, which is necessary for the surface area calculation.
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Integration

Integration is a fundamental concept in calculus used to find areas, volumes, and other quantities. In this context, it involves evaluating the integral of the surface area formula over the specified limits (from y = 2 to y = 10). Mastery of definite integrals is essential for accurately computing the total surface area generated by the revolution of the curve.
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Related Practice
Textbook Question

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


y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis 

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Textbook Question

Explain how to find the mass of a one-dimensional object with a variable density ρ.

Textbook Question

What is the pressure on a horizontal surface with an area of 2 m² that is 4 m underwater?

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Textbook Question

Use the general slicing method to find the volume of the following solids.

The solid whose base is the triangle with vertices (0, 0), (2, 0), and (0, 2), and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles

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Textbook Question

Determine the area of the shaded region in the following figures.

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Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=2x,y=0 , and x=3; about the x-axis (Verify that your answer agrees with the volume formula for a cone.)