a) Explanation Given: The radius of the sphere is R . The distance between the planes is h . Calculation: Consider the equation for sphere as follows: x 2 + y 2 = R 2 (1) Rearrange Equation (1). y 2 = R 2 − x 2 y = R 2 − x 2 (2) Draw the graph as shown in Figure 1: Differentiate both sides of the Equation (2). d y d x = 1 2 ( R 2 − x 2 ) − 1 2 ( − 2 x ) = − x ( R 2 − x 2 ) 1 2 = − x R 2 − x 2 Calculate the ds using formula: d s = 1 + ( d y d x ) 2 d x (3) Substitute ( − x R 2 − x 2 ) for d x d y in Equation (3). d s = 1 + ( − x R 2 − x 2 ) 2 d x = 1 + x 2 R 2 − x 2 d x Calculate the surface area of a zone of a sphere using the formula: S = ∫ a b 2 π y d s (4) Substitute ( a + h ) for b , R 2 − x 2 for y , and 1 + x 2 R 2 − x 2 d x for ds in Equation (4) b) To determine To prove: the surface area of a zone of a cylinder is same as the surface area of zone of a sphere in part (a).

8th Edition

ISBN: 9781305266636

Author: James Stewart

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 8.2, Problem 36E

To determine

To prove: the surface area of a zone of a sphere that lies between two parallel planes is S=2πRh.

To determine

To prove: the surface area of a zone of a cylinder is same as the surface area of zone of a sphere in part (a).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.2, Problem 35E

Chapter 8.2, Problem 37E

Students have asked these similar questions

Question 1. (10 points) A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by dV = 1.45V(2 In(V+1)). dt (a) (4 pts) Find all the equilibria and determine their stability using the stability condition. (b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable. (c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain in biological terms what happens to the size of each of these tumours at time progresses.

For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.

Question 2: (10 points) Evaluate the definite integral. Use the following form of the definition of the integral to evaluate the integral: Theorem: Iff is integrable on [a, b], then where Ax = (ba)/n and x₂ = a + i^x. You might need the following formulas. IM³ L² (3x² (3x²+2x- 2x - 1)dx. n [f(z)dz lim f(x)Az a n→∞ i=1 n(n + 1) 2 n i=1 n(n+1)(2n+1) 6

Chapter 8 Solutions

Single Variable Calculus

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

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.

To prove : the surface area of a zone of a sphere that lies between two parallel planes is S = 2 π R h .

Solution Summary: The author explains that the surface area of a sphere that lies between two parallel planes is proved.