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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 6.4.72b
Chapter 6, Problem 6.4.72b

Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)


b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

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Step 1: Observe the integral on the left-hand side, ∫₂₀(25−(x²+1)²) dx. This represents the area under the curve defined by the function f(x) = 25−(x²+1)² over the interval [0, 2]. The function is symmetric about the y-axis, and its shape suggests a geometric interpretation.
Step 2: Rewrite the integral on the right-hand side, 2∫₁⁵ y√y−1 dy. This represents twice the area under the curve defined by g(y) = y√y−1 over the interval [1, 5]. The factor of 2 indicates symmetry or a doubling of the area.
Step 3: Use the hint to draw pictures. For the left-hand side, sketch the curve f(x) = 25−(x²+1)², which is a parabola-like shape inverted due to the negative sign. For the right-hand side, sketch the curve g(y) = y√y−1, which is a function involving a square root and grows as y increases.
Step 4: Analyze the symmetry and transformations. The integral on the left-hand side can be interpreted geometrically as the area of a region that corresponds to the integral on the right-hand side after a change of variables. Specifically, the substitution x²+1 ↔ y and adjustments to limits of integration align the two integrals.
Step 5: Conclude that the equality holds because the two integrals represent the same geometric area under their respective curves, albeit expressed in different coordinate systems. The factor of 2 in the right-hand side accounts for symmetry or doubling of the area.

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Key Concepts

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

Definite Integrals

Definite integrals represent the signed area under a curve between two points on the x-axis. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Understanding the properties of definite integrals, such as linearity and symmetry, is crucial for analyzing equalities involving integrals without direct evaluation.
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Substitution Method

The substitution method is a technique used in integration to simplify the process by changing variables. This method involves substituting a part of the integrand with a new variable, which can make the integral easier to evaluate. Recognizing when and how to apply substitution is essential for understanding the relationship between different integrals, as seen in the given equality.
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Geometric Interpretation of Integrals

Integrals can be interpreted geometrically as areas under curves or between curves. Visualizing the functions involved in the integrals can reveal relationships and symmetries that may not be immediately apparent through algebraic manipulation. Drawing diagrams helps in understanding how the areas represented by the integrals relate to each other, which is key to explaining the given equality.
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Related Practice
Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point x in [0, 4]?

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Textbook Question

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


b. What is the SAV ratio of a ball with radius a? 

Textbook Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


b. What is the inner radius of a cross section of the solid at a point y in [1, 3]?

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Textbook Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


b. Repeat part (a) using the disk method.

Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

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Textbook Question

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


b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.