1. In this workshop, we are going to explore formulas for areas and arc length associated to curves writtenin polar coordinates. We'll do this by using the formulas to compute areas and lengths of well-knownobjects (and, in most cases, checking our answers).Here is the formula for the area bounded by r = f(0) and the rays 0 = a and 0 = ß:A :(S(C(a) Consider the circle of radius 2 centered at the origin. Write down an integral in polar coordinatesthat gives the area inside the circle. Evaluate your integral, and check that you get the correctanswer. What would change if you only wanted to find the area of the quarter-circle in the secondquadrant?(b) Now, let's make our circle of radius 2 not centered at the origin, but instead at (2, 0). What isr = f(0) in this case? (If you were paying attention, we did this in our last class.) Write down anintegral in polar coordinates that gives the area inside the circle. (Be careful with your bounds.)Evaluate your integral, and check that you get the correct answer.How about other shapes?(c) Consider the triangle with vertices at (0,0), (5,0), and (5,5). Write down an integral in polarcoordinates that gives the area of this triangle. (You can do this with exactly one integral.) Evaluateyour integral, and check that you get the correct answer (you should know what the correct answeris).(d) Consider the rectangular region defined by 1 < r < 3 and 0 < y < 2. Write down one or moreintegrals in polar coordinates to find the area of the region. (You may need to add or subtract twoor more integrals to achieve this.) You do not need to evaluate these integrals.We also have a formula for the arc length of a curve r = {(0) defined in polar coordinates:s = /' vt@)² + (s"(0)² d0 (e) Write down an integral for the circumference of the circle of radius 2 centered at the origin. Evaluateyour integral, and check that you get the correct answer.(f) Now, let's make our circle of radius 2 not centered at the origin, but instead at (2,0). Write downan integral in polar coordinates that gives the circumference of the circle. (Be careful with yourbounds.) Evaluate your integral, and check that you get the correct answer.(g) Consider the line segment from (1, 1) to (0, 1). Write down an integral that computes the length ofthis line segment, using polar coordinates. Evaluate your integral, and check that you obtain thecorrect answer. (You'll need to use some trigonometric identities here.)

Since you have asked for a multiple question, we will the first question for you. If you want any…

1. In this workshop, we are going to explore formulas for areas and arc length associated to curves written in polar coordinates. We'll do this by using the formulas to compute areas and lengths of well-known objects (and, in most cases, checking our answers). Here is the formula for the area bounded by r = f(0) and the rays 0 = a and 0 = ß: (a) Consider the circle of radius 2 centered at the origin. Write down an integral in polar coordinates that gives the area inside the circle. Evaluate your integral, and check that you get the correct answer. What would change if you only wanted to find the area of the quarter-circle in the second quadrant?

1. In this workshop, we are going to explore formulas for areas and arc length associated to curves written in polar coordinates. We'll do this by using the formulas to compute areas and lengths of well-known objects (and, in most cases, checking our answers). Here is the formula for the area bounded by r = f(0) and the rays 0 = a and 0 = ß: (a) Consider the circle of radius 2 centered at the origin. Write down an integral in polar coordinates that gives the area inside the circle. Evaluate your integral, and check that you get the correct answer. What would change if you only wanted to find the area of the quarter-circle in the second quadrant?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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 selection of the best element for a given system.

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, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

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 that have length, breadth, and height (or depth).

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.

Question

1. In this workshop, we are going to explore formulas for areas and arc length associated to curves written
in polar coordinates. We'll do this by using the formulas to compute areas and lengths of well-known
objects (and, in most cases, checking our answers).
Here is the formula for the area bounded by r = f(0) and the rays 0 = a and 0 = ß:
A :
(S(C
(a) Consider the circle of radius 2 centered at the origin. Write down an integral in polar coordinates
that gives the area inside the circle. Evaluate your integral, and check that you get the correct
answer. What would change if you only wanted to find the area of the quarter-circle in the second
quadrant?
(b) Now, let's make our circle of radius 2 not centered at the origin, but instead at (2, 0). What is
r = f(0) in this case? (If you were paying attention, we did this in our last class.) Write down an
integral in polar coordinates that gives the area inside the circle. (Be careful with your bounds.)
Evaluate your integral, and check that you get the correct answer.
How about other shapes?
(c) Consider the triangle with vertices at (0,0), (5,0), and (5,5). Write down an integral in polar
coordinates that gives the area of this triangle. (You can do this with exactly one integral.) Evaluate
your integral, and check that you get the correct answer (you should know what the correct answer
is).
(d) Consider the rectangular region defined by 1 < r < 3 and 0 < y < 2. Write down one or more
integrals in polar coordinates to find the area of the region. (You may need to add or subtract two
or more integrals to achieve this.) You do not need to evaluate these integrals.
We also have a formula for the arc length of a curve r = {(0) defined in polar coordinates:
s = /' vt@)² + (s"(0)² d0

(e) Write down an integral for the circumference of the circle of radius 2 centered at the origin. Evaluate
your integral, and check that you get the correct answer.
(f) Now, let's make our circle of radius 2 not centered at the origin, but instead at (2,0). Write down
an integral in polar coordinates that gives the circumference of the circle. (Be careful with your
bounds.) Evaluate your integral, and check that you get the correct answer.
(g) Consider the line segment from (1, 1) to (0, 1). Write down an integral that computes the length of
this line segment, using polar coordinates. Evaluate your integral, and check that you obtain the
correct answer. (You'll need to use some trigonometric identities here.)

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