Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explores the use of slicing and Riemann sums to compute areas and volumes of various shapes including semicircles, cones, hemispheres, and pyramids. By partitioning the region into manageable slices (either horizontal or vertical), expressing dimensions in terms of a common variable, and then summing these contributions via a definite integral, we can derive fundamental formulas and solve complex geometry problems. Key techniques include applying the Pythagorean theorem, using similar triangles, and understanding the role of limits in transitioning from sums to integrals.

Learning Objectives

1

Explain how to calculate areas and volumes using the method of slicing and Riemann sums.

2

Set up definite integrals by identifying proper limits and slice dimensions in geometric shapes.

3

Use horizontal slicing to derive area formulas for regions, such as semicircles, and volume formulas for solids like cones, hemispheres, and pyramids.

4

Apply the Pythagorean theorem to relate variables and dimensions in setting up integrals.

5

Evaluate definite integrals to obtain exact or approximate measures for areas and volumes.

Key Concepts

CONCEPT

DEFINITION

Riemann Sum

An approximation of the total area or volume by summing contributions of small, discrete slices or pieces, which approaches the definite integral as the slice thickness tends to zero.

Definite Integral

The limit of a Riemann sum as the partition becomes infinitely fine, representing the exact accumulation of quantities such as area or volume over an interval.

Slicing Method

A technique that divides a region or solid into thin slices (disks, strips, or shells) to approximate areas or volumes and then sums them via integration.

Horizontal Slice

A slice cut parallel to the base of a shape, often used when the variable of integration is the vertical height (h).

Pythagorean Theorem

A fundamental relation in Euclidean geometry among the three sides of a right triangle, used to relate dimensions in problem setups involving circles and triangles.

Example Problems

Example 1

(a) Write a Riemann sum approximating the area of the region in Figure $8.13,$ using vertical strips as shown. (b) Evaluate the corresponding integral. (FIGURE CAN'T COPY)

Example 2

(a) Write a Riemann sum approximating the area of the region in Figure $8.14,$ using vertical strips as shown. (b) Evaluate the corresponding integral. (FIGURE CAN'T COPY)

Example 3

(a) Write a Riemann sum approximating the area of the region in Figure $8.15,$ using horizontal strips as shown. (b) Evaluate the corresponding integral. (FIGURE CAN'T COPY)

Example 4

(a) Write a Riemann sum approximating the area of the region in Figure $8.16,$ using horizontal strips as shown. (b) Evaluate the corresponding integral. (FIGURE CAN'T COPY)

Example 5

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly. (FIGURE CAN'T COPY)
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Step-by-Step Explanations

QUESTION

How can we set up and evaluate a definite integral to determine the area of a semicircle of radius 7 cm using horizontal slices?

STEP-BY-STEP ANSWER:

Step 1: Recognize that a horizontal slice at height h has a thickness Δh and a width given by the chord length at that height.
Step 2: Use the Pythagorean theorem. For a semicircle with radius 7, at height h the half-width is √(49 − h²), so the full width is 2√(49 − h²).
Step 3: Write the approximate area for a slice as Area_slice ≈ 2√(49 − h²)Δh.
Step 4: Sum over all slices from h = 0 to h = 7, yielding the Riemann sum that approaches the integral: Area = 2 ∫_0^7 √(49 − h²) dh.
Step 5: Evaluate the integral using standard formulas, obtaining the area of the semicircle.
Final Answer:

Area of a Semicircle by Horizontal Slices

QUESTION

How can horizontal slicing be used to set up a definite integral to find the volume of a cone with a given height and base dimensions?

STEP-BY-STEP ANSWER:

Step 1: Divide the cone horizontally into thin, circular slices of thickness Δh.
Step 2: At a height h above the base, determine the radius of the slice using similar triangles. For a cone with height H and base diameter such that width w = H − 2h, the radius becomes (constant factor) × (H/2 − h).
Step 3: Express the volume of a slice as Volume_slice ≈ π (radius)² Δh.
Step 4: Sum over all slices from h = 0 to h = H, forming the Riemann sum: Volume = ∫_0^H π (radius expressed in terms of h)² dh.
Step 5: Evaluate the integral to find the exact volume of the cone.
Final Answer:

Volume of a Cone by Horizontal Slicing

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Common Mistakes

  • Choosing incorrect limits of integration by not matching the variable with the actual dimensions of the slices.
  • Mixing up horizontal and vertical slicing strategies, leading to incorrect expressions for the dimensions.
  • Forgetting to correctly express geometric quantities (such as width or radius) in terms of the variable of integration.
  • Omitting the necessary factors (like 2 in the width of horizontal slices) in the setup of Riemann sums.
  • Overlooking the proper use of similar triangles or the Pythagorean theorem to relate dimensions in geometric figures.