(a) Explanation Given: The solid region: Formula used: The volume of the solid in cylindrical coordinates is: V = ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The rectangular conversion equations for the cylindrical coordinates are: x = r cos θ y = r sin θ z = z Calculation: The limits of r , θ and z for the bounded solid is as follows: The limits for z are: 2 ≤ z ≤ 8 − x 2 − y 2 2 ≤ z ≤ 8 − r 2 The limits for r are: 0 ≤ r ≤ 2 The limits for θ are: 0 ≤ θ ≤ 2 π Put the limits of r , θ and z in the formula for the volume, V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ Solve the above integral to get the volume of the bounded solid, V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ = ∫ 0 2 π ∫ 0 2 ( ∫ 2 8 − r 2 r d z ) d r d θ = ∫ 0 2 π ∫ 0 2 ( r [ z ] 2 8 − r 2 ) d r d θ = ∫ 0 2 π ∫ 0 2 r ( 8 − r 2 − 2 ) d r d θ On further simplification, V = ∫ 0 2 π ∫ 0 2 r 8 − r 2 d r d θ − ∫ 0 2 π ∫ 0 2 2 r d r d θ = ∫ 0 2 π ( ∫ 0 2 (b) To determine To calculate: The volume of the solid bounded below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using spherical coordinates.

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 14, Problem 18PS

(a)

To determine

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

(b)

To determine

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14, Problem 17PS

Chapter 15.1, Problem 65E

Students have asked these similar questions

A body of mass m at the top of a 100 m high tower is thrown vertically upward with an initial velocity of 10 m/s. Assume that the air resistance FD acting on the body is proportional to the velocity V, so that FD=kV. Taking g = 9.75 m/s2 and k/m = 5 s, determine: a) what height the body will reach at the top of the tower, b) how long it will take the body to touch the ground, and c) the velocity of the body when it touches the ground.

A chemical reaction involving the interaction of two substances A and B to form a new compound X is called a second order reaction. In such cases it is observed that the rate of reaction (or the rate at which the new compound is formed) is proportional to the product of the remaining amounts of the two original substances. If a molecule of A and a molecule of B combine to form a molecule of X (i.e., the reaction equation is A + B ⮕ X), then the differential equation describing this specific reaction can be expressed as: dx/dt = k(a-x)(b-x) where k is a positive constant, a and b are the initial concentrations of the reactants A and B, respectively, and x(t) is the concentration of the new compound at any time t. Assuming that no amount of compound X is present at the start, obtain a relationship for x(t). What happens when t ⮕∞?

Consider a body of mass m dropped from rest at t = 0. The body falls under the influence of gravity, and the air resistance FD opposing the motion is assumed to be proportional to the square of the velocity, so that FD = kV2. Call x the vertical distance and take the positive direction of the x-axis downward, with origin at the initial position of the body. Obtain relationships for the velocity and position of the body as a function of time t.

Chapter 14 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrd Ch. 14.1 - Prob. 1E Ch. 14.1 - Evaluate the integral: xx2yxdy Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Evaluate the integral: 04x2x2ydy Ch. 14.1 - Evaluate the integral: x3x(x2+3y2)dy Ch. 14.1 - Evaluate the integral: eyyylnxxdx;y0 Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dx Ch. 14.1 - Evaluate the integral: 0x2yeyxdy

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 bounded below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using cylindrical coordinates.

Solution Summary: The author explains how to calculate the volume of the solid bounded below by the plane and above the sphere using cylindrical coordinates.

Videos

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

Want to see the full answer?

Chapter 14 Solutions