Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.4.4

Chapter 6, Problem 6.4.4

Look again at the region R in Figure 6.38 (p. 439). Explain why it would be difficult to use the washer method to find the volume of the solid of revolution that results when R is revolved about the y-axis.

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Identify the region R and the axis of revolution, which is the y-axis in this case.

Recall that the washer method involves slicing the solid perpendicular to the axis of revolution, creating washers (disks with holes) whose volume can be integrated.

When revolving around the y-axis, the slices should be horizontal (parallel to the x-axis) to form washers with radii expressed as functions of y.

If the region R is bounded by curves that are easier to express as functions of x rather than y, it becomes difficult to describe the inner and outer radii of the washers in terms of y, complicating the integral setup.

Therefore, the difficulty arises because the boundaries of R may not be easily invertible or expressible as functions of y, making the washer method cumbersome for this axis of revolution.

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Key Concepts

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

Washer Method for Volumes of Revolution

The washer method involves slicing a solid perpendicular to the axis of revolution, creating washers (disks with holes). It requires expressing the outer and inner radii as functions of the variable of integration, typically aligned with the axis of revolution.

Axis of Revolution and Variable of Integration

When revolving a region about the y-axis, the variable of integration is usually y. To use the washer method effectively, the boundaries of the region must be easily expressed as functions of y, which can be challenging if the region is defined primarily in terms of x.

Complexity of Expressing Radii in Terms of y

If the region R is bounded by curves given as functions of x, rewriting these boundaries as functions of y may be difficult or impossible to do explicitly. This complicates finding the inner and outer radii for washers, making the washer method less practical.

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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=4−x^2,x=2, and y=4; about the y-axis

141

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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 = x³ ,y = 1, and x = 0; about the x-axis

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Textbook Question

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = −32; v(0)=50; s(0)=0

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Textbook Question

Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.

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Textbook Question

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

The region bounded by y=√x, y=2x−15, and y=0

109

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Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.

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