Math 152 Week 15 Workshop Problems 4/22/24 Write up your solutions to each of these problems on a seperate sheet of paper. 1. Consider the polar equation r = 1 - 2 sin(0), for 0≤0≤2π (a limacon). (a) Graph this equation as a Cartesian equation in the re- plane. (Your graph should look like a sine wave with some transformations done to it.) (b) Using your graph in part a) as a guide, sketch the graph of the above equation as a polar equation in the xy-plane. Then use a graphing utility to check your work. (c) Set up, but do not evaluate, a polar integral to calculate the area of the inner loop of the limacon. 2. Consider the two curves r₁ = 2 sin(0) and r2 = 2 sin(0) and r2 = 2√3 cos(0). r1 (a) Graph the two curves in the same xy-plane. (b) On your graph, shade the region in the first quadrant that is outside r₁ and inside r2. Then set up an integral to find its area. (c) On your graph, shade (with a different color or different pattern) the region in the first quadrant that is outside r2 and inside r₁. Then, set up an integral to find its area. 3. Consider the polar curve r = 2+2 cos(0). (a) Show that the Cartesian coordinate (0, 2) lies on this curve by converting the point into polar and showing that the polar coordinate satisfies the above equation. (b) Find the Cartesian equation of the line tangent to the polar curve at this point. (c) Find the arc length of this curve on the interval 0 ≤ 0≤πT.
Math 152 Week 15 Workshop Problems 4/22/24 Write up your solutions to each of these problems on a seperate sheet of paper. 1. Consider the polar equation r = 1 - 2 sin(0), for 0≤0≤2π (a limacon). (a) Graph this equation as a Cartesian equation in the re- plane. (Your graph should look like a sine wave with some transformations done to it.) (b) Using your graph in part a) as a guide, sketch the graph of the above equation as a polar equation in the xy-plane. Then use a graphing utility to check your work. (c) Set up, but do not evaluate, a polar integral to calculate the area of the inner loop of the limacon. 2. Consider the two curves r₁ = 2 sin(0) and r2 = 2 sin(0) and r2 = 2√3 cos(0). r1 (a) Graph the two curves in the same xy-plane. (b) On your graph, shade the region in the first quadrant that is outside r₁ and inside r2. Then set up an integral to find its area. (c) On your graph, shade (with a different color or different pattern) the region in the first quadrant that is outside r2 and inside r₁. Then, set up an integral to find its area. 3. Consider the polar curve r = 2+2 cos(0). (a) Show that the Cartesian coordinate (0, 2) lies on this curve by converting the point into polar and showing that the polar coordinate satisfies the above equation. (b) Find the Cartesian equation of the line tangent to the polar curve at this point. (c) Find the arc length of this curve on the interval 0 ≤ 0≤πT.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Q1 please
Transcribed Image Text:Math 152
Week 15 Workshop Problems
4/22/24
Write up your solutions to each of these problems on a seperate sheet of paper.
1. Consider the polar equation r = 1 - 2 sin(0), for 0≤0≤2π (a limacon).
(a) Graph this equation as a Cartesian equation in the re- plane. (Your graph should look like a
sine wave with some transformations done to it.)
(b) Using your graph in part a) as a guide, sketch the graph of the above equation as a polar equation
in the xy-plane. Then use a graphing utility to check your work.
(c) Set up, but do not evaluate, a polar integral to calculate the area of the inner loop of the limacon.
2. Consider the two curves r₁ = 2 sin(0) and r2 =
2 sin(0) and r2 = 2√3 cos(0).
r1
(a) Graph the two curves in the same xy-plane.
(b) On your graph, shade the region in the first quadrant that is outside r₁ and inside r2. Then set
up an integral to find its area.
(c) On your graph, shade (with a different color or different pattern) the region in the first quadrant
that is outside r2 and inside r₁. Then, set up an integral to find its area.
3. Consider the polar curve r = 2+2 cos(0).
(a) Show that the Cartesian coordinate (0, 2) lies on this curve by converting the point into polar and
showing that the polar coordinate satisfies the above equation.
(b) Find the Cartesian equation of the line tangent to the polar curve at this point.
(c) Find the arc length of this curve on the interval 0 ≤ 0≤πT.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning