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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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
Transcribed Image Text: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.)
Transcribed Image Text:(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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