(a) Explanation Given: The surface f ( x , y ) = 16 − x 2 − y 2 . The elliptical region x 2 16 + y 2 9 ≤ 1 . For the ellipse x 2 a 2 + y 2 b 2 = 1 , the transformations are x = a u and y = b v . Formula used: If f ( x , y ) is integrable over a plane region R and f ( x , y ) ≥ 0 for all ( x , y ) in R , then the volume of the solid region that lies above R and below the graph of f ( x , y ) is, V = ∬ R f ( x , y ) d A Integration property, ∫ cos a x d x = sin a x a + C And, ∫ x n d x = x n + 1 n + 1 + C Trigonometric property, cos 2 x = 1 2 ( 1 + cos 2 x ) sin 2 x = 1 2 ( 1 − cos 2 x ) Calculation: Consider the surface, f ( x , y ) = 16 − x 2 − y 2 . And, the region x 2 16 + y 2 9 ≤ 1 . Rewrite the above equation as, x 2 4 2 + y 2 3 2 ≤ 1 From the given information, the transformations are x = 4 u and y = 3 v . Therefore, the image equation for elliptical region is, ( 4 u ) 2 4 2 + ( 3 v ) 2 3 2 ≤ 1 u 2 + v 2 ≤ 1 For the limits, The equation for image is a standard equation of circle with the center at origin on the uv -plane with radius 1. So, the graph of the image S is, Therefore, the region is bounded by − 1 ≤ v ≤ 1 . Now, write the image equation in terms of u to find the limit of u , u 2 + v 2 = 1 u 2 = 1 − v 2 u = ± 1 − v 2 Therefore, the limit of u is − 1 − v 2 ≤ u ≤ 1 − v 2 . Thus, the volume of the solid is, V = ∫ − 1 1 ∫ − 1 − v 2 1 − v 2 ( 16 − ( 4 u ) 2 − ( 3 v ) 2 ) d ( 4 u ) d ( 3 v ) = ∫ − 1 1 ∫ − 1 − v 2 1 − v 2 ( 16 − 16 u 2 − 9 v 2 ) 4 × 3 d u d v = ∫ − 1 1 ∫ − 1 − v 2 1 − v 2 ( 16 − 16 u 2 − 9 v 2 ) 12 d u d v Assume u = r cos θ and v = r sin θ . And, it is known that area d A = r d r d θ in polar form. Since, it is a circle of radius 1 therefore the limit of r will be 0 ≤ r ≤ 1 . And, the limit of θ will be 0 ≤ θ ≤ 2 π (b) To determine To calculate: The volume of the solid which lies below the surface f ( x , y ) = A cos ( π 2 x 2 a 2 + y 2 b 2 ) and above the elliptical region x 2 a 2 + y 2 b 2 ≤ 1 .

10th Edition

ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 14.8, Problem 32E

(a)

To determine

To calculate: The volume of the solid which lies below the surface f(x,y)=16−x2−y2 and above the elliptical region x216+y29≤1.

(b)

To determine

To calculate: The volume of the solid which lies below the surface f(x,y)=Acos(π2x2a2+y2b2) and above the elliptical region x2a2+y2b2≤1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.8, Problem 31E

Chapter 14.8, Problem 33E

Students have asked these similar questions

A cable runs along the wall from C to P at a cost of $24 per meter, and straight from P to M at a cost of $26 per meter. If M is 10 meters from the nearest point A on the wall where P lies, and A is 72 meters from C, find the distance from C to P such that the cost of installing the cable is minimized and find this cost. C 72 P A 10 M

The number of bank robberies in a country for the years 2010-2018 is given in the following figure. Consider the closed interval [2010,2018]. (a) Give all relative maxima and minima and when they occur on the interval. (b) Give the absolute maxima and minima and when they occur on the interval. Incidents 7000- 6000-5 5482 5000- 4424 4273 4822 4000- 3708 3748 4229 4089 3000- 2582 2000- 1000- 0 2010 2012 2014 2016 2018 Year

please do 8.1 q7

Chapter 14 Solutions

Calculus

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The volume of the solid which lies below the surface f ( x , y ) = 16 − x 2 − y 2 and above the elliptical region x 2 16 + y 2 9 ≤ 1 .

Solution Summary: The author calculates the volume of the solid below the surface f(x,y) and above the elliptical region.