Calculus: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134763644
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Chapter 12.1, Problem 110E
To determine
The surface area of the curve
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Surfaces of revolution Suppose y = ƒ(x) is a continuous and positive function on [a, b]. Let S be the surface generated when the graph of ƒ on [a, b] is revolved about the x-axis.a. Show that S is described parametrically byr(u, v) = ⟨u, ƒ(u) cos v, ƒ(u) sin v⟩, for a ≤ u ≤ b, 0 ≤ v ≤ 2π.b. Find an integral that gives the surface area of S.c. Apply the result of part (b) to the surface generated withƒ(x) = x3, for 1 ≤ x ≤ 2.d. Apply the result of part (b) to the surface generated withƒ(x) = (25 - x2)1/2, for 3 ≤ x ≤ 4.
Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane.
a.In each case, find z’(t).
b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing).
z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2π
Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane.
a. In each case, find z'(t).
b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing).
z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2π
Chapter 12 Solutions
Calculus: Early Transcendentals (3rd Edition)
Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QCCh. 12.1 - Describe the curve generated by x = 3 + 2t, y = 12...Ch. 12.1 - Find parametric equations for the line segment...Ch. 12.1 - Use Theorem 12.1 to find the slope of the line x =...Ch. 12.1 - Use the arc length formula to find the length of...Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2ECh. 12.1 - Prob. 3ECh. 12.1 - Give parametric equations that generate the line...
Ch. 12.1 - Find parametric equations for the complete...Ch. 12.1 - Describe the similarities between the graphs of...Ch. 12.1 - Find the slope of the parametric curve x = 2t3 +...Ch. 12.1 - Prob. 8ECh. 12.1 - Find three different pairs of parametric equations...Ch. 12.1 - Use calculus to find the arc length of the line...Ch. 12.1 - Prob. 11ECh. 12.1 - Prob. 12ECh. 12.1 - Prob. 13ECh. 12.1 - Prob. 14ECh. 12.1 - Prob. 15ECh. 12.1 - Prob. 16ECh. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - Prob. 21ECh. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - Prob. 27ECh. 12.1 - Working with parametric equations Consider the...Ch. 12.1 - Prob. 29ECh. 12.1 - Prob. 30ECh. 12.1 - Eliminating the parameter Eliminate the parameter...Ch. 12.1 - Eliminating the parameter Eliminate the parameter...Ch. 12.1 - Prob. 33ECh. 12.1 - Prob. 34ECh. 12.1 - Prob. 35ECh. 12.1 - Prob. 36ECh. 12.1 - Parametric equations of circles Find parametric...Ch. 12.1 - Parametric equations of circles Find parametric...Ch. 12.1 - Parametric equations of circles Find parametric...Ch. 12.1 - Parametric equations of circles Find parametric...Ch. 12.1 - Prob. 41ECh. 12.1 - Curves to parametric equations Find parametric...Ch. 12.1 - Curves to parametric equations Give a set of...Ch. 12.1 - Curves to parametric equations Give a set of...Ch. 12.1 - Prob. 45ECh. 12.1 - Curves to parametric equations Find parametric...Ch. 12.1 - Prob. 47ECh. 12.1 - Prob. 48ECh. 12.1 - Prob. 49ECh. 12.1 - Curves to parametric equations Find parametric...Ch. 12.1 - Curves to parametric equations Find parametric...Ch. 12.1 - Curves to parametric equations Find parametric...Ch. 12.1 - Circular motion Find parametric equations that...Ch. 12.1 - Circular motion Find parametric equations that...Ch. 12.1 - Circular motion Find parametric equations that...Ch. 12.1 - Circular motion Find parametric equations that...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - More parametric curves Use a graphing utility to...Ch. 12.1 - Implicit function graph Explain and carry out a...Ch. 12.1 - Air drop A plane traveling horizontally at 80 m/s...Ch. 12.1 - Air dropinverse problem A plane traveling...Ch. 12.1 - Prob. 66ECh. 12.1 - Prob. 67ECh. 12.1 - Derivatives Consider the following parametric...Ch. 12.1 - Derivatives Consider the following parametric...Ch. 12.1 - Prob. 70ECh. 12.1 - Derivatives Consider the following parametric...Ch. 12.1 - Prob. 72ECh. 12.1 - Tangent lines Find an equation of the line tangent...Ch. 12.1 - Tangent lines Find an equation of the line tangent...Ch. 12.1 - Tangent lines Find an equation of the line tangent...Ch. 12.1 - Tangent lines Find an equation of the line tangent...Ch. 12.1 - Slopes of tangent lines Find all the points at...Ch. 12.1 - Slopes of tangent lines Find all the points at...Ch. 12.1 - Slopes of tangent lines Find all the points at...Ch. 12.1 - Slopes of tangent lines Find all the points at...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Arc length Find the arc length of the following...Ch. 12.1 - Explain why or why not Determine whether the...Ch. 12.1 - Prob. 90ECh. 12.1 - Prob. 91ECh. 12.1 - Prob. 92ECh. 12.1 - Parametric equations of ellipses Find parametric...Ch. 12.1 - Prob. 94ECh. 12.1 - Prob. 95ECh. 12.1 - Prob. 96ECh. 12.1 - Prob. 97ECh. 12.1 - Beautiful curves Consider the family of curves...Ch. 12.1 - Prob. 99ECh. 12.1 - Prob. 100ECh. 12.1 - Prob. 101ECh. 12.1 - Lissajous curves Consider the following Lissajous...Ch. 12.1 - Area under a curve Suppose the function y = h(x)...Ch. 12.1 - Area under a curve Suppose the function y = h(x)...Ch. 12.1 - Area under a curve Suppose the function y = h(x)...Ch. 12.1 - Prob. 106ECh. 12.1 - Prob. 107ECh. 12.1 - Prob. 108ECh. 12.1 - Surfaces of revolution Let C be the curve x =...Ch. 12.1 - Prob. 110ECh. 12.1 - Surfaces of revolution Let C be the curve x =...Ch. 12.1 - Prob. 112ECh. 12.1 - Prob. 113ECh. 12.1 - Prob. 114ECh. 12.2 - Which of the following coordinates represent the...Ch. 12.2 - Draw versions of Figure 12.21 with P in the...Ch. 12.2 - Give two polar coordinate descriptions of the...Ch. 12.2 - Describe the polar curves r = 12, r = 6, and r sin...Ch. 12.2 - Prob. 5QCCh. 12.2 - Prob. 6QCCh. 12.2 - Plot the points with polar coordinates (2,6) and...Ch. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - What is the polar equation of the vertical line x...Ch. 12.2 - What is the polar equation of the horizontal line...Ch. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Graph the points with the following polar...Ch. 12.2 - Graph the points with the following polar...Ch. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Points in polar coordinates Give two sets of polar...Ch. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Prob. 20ECh. 12.2 - Prob. 21ECh. 12.2 - Prob. 22ECh. 12.2 - Rader Airplanes are equipped with transponders...Ch. 12.2 - Prob. 24ECh. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following polar...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Converting coordinates Express the following...Ch. 12.2 - Prob. 37ECh. 12.2 - Prob. 38ECh. 12.2 - Prob. 39ECh. 12.2 - Prob. 40ECh. 12.2 - Prob. 41ECh. 12.2 - Prob. 42ECh. 12.2 - Prob. 43ECh. 12.2 - Prob. 44ECh. 12.2 - Prob. 45ECh. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Cartesian-to-polar coordinates Convert the...Ch. 12.2 - Cartesian-to-polar coordinates Convert the...Ch. 12.2 - Cartesian-to-polar coordinates Convert the...Ch. 12.2 - Cartesian-to-polar coordinates Convert the...Ch. 12.2 - Prob. 53ECh. 12.2 - Prob. 54ECh. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Graphing polar curves Graph the following...Ch. 12.2 - Prob. 65ECh. 12.2 - Prob. 66ECh. 12.2 - Prob. 67ECh. 12.2 - Prob. 68ECh. 12.2 - Using a graphing utility Use a graphing utility to...Ch. 12.2 - Using a graphing utility Use a graphing utility to...Ch. 12.2 - Prob. 71ECh. 12.2 - Using a graphing utility Use a graphing utility to...Ch. 12.2 - Using a graphing utility Use a graphing utility to...Ch. 12.2 - Using a graphing utility Use a graphing utility to...Ch. 12.2 - Prob. 75ECh. 12.2 - Prob. 76ECh. 12.2 - Prob. 77ECh. 12.2 - Prob. 78ECh. 12.2 - Circles in general Show that the polar equation...Ch. 12.2 - Prob. 80ECh. 12.2 - Prob. 81ECh. 12.2 - Prob. 82ECh. 12.2 - Prob. 83ECh. 12.2 - Equations of circles Find equations of the circles...Ch. 12.2 - Navigating A plane is 150 miles north of a radar...Ch. 12.2 - Prob. 86ECh. 12.2 - Prob. 87ECh. 12.2 - Prob. 88ECh. 12.2 - Prob. 89ECh. 12.2 - Prob. 90ECh. 12.2 - Prob. 91ECh. 12.2 - Limiting limaon Consider the family of limaons r =...Ch. 12.2 - Prob. 93ECh. 12.2 - Prob. 94ECh. 12.2 - Prob. 95ECh. 12.2 - The lemniscate family Equations of the form r2 = a...Ch. 12.2 - The rose family Equations of the form r = a sin m...Ch. 12.2 - Prob. 98ECh. 12.2 - Prob. 99ECh. 12.2 - The rose family Equations of the form r = a sin m...Ch. 12.2 - Prob. 101ECh. 12.2 - Prob. 102ECh. 12.2 - Prob. 103ECh. 12.2 - Spirals Graph the following spirals. Indicate the...Ch. 12.2 - Enhanced butterfly curve The butterfly curve of...Ch. 12.2 - Prob. 106ECh. 12.2 - Prob. 107ECh. 12.2 - Prob. 108ECh. 12.2 - Prob. 109ECh. 12.2 - Prob. 110ECh. 12.2 - Cartesian lemniscate Find the equation in...Ch. 12.3 - Verify that if y = f() sin , then y'() =f'() sin ...Ch. 12.3 - Prob. 2QCCh. 12.3 - Prob. 3QCCh. 12.3 - Prob. 4QCCh. 12.3 - Prob. 1ECh. 12.3 - Explain why the slope of the line = /2 is...Ch. 12.3 - Explain why the slope of the line tangent to the...Ch. 12.3 - What integral must be evaluated to find the area...Ch. 12.3 - What is the slope of the line = /3?Ch. 12.3 - Prob. 6ECh. 12.3 - Find the area of the shaded region.Ch. 12.3 - Prob. 8ECh. 12.3 - Explain why the point with polar coordinates (0,...Ch. 12.3 - Prob. 10ECh. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Slopes of tangent lines Find the slope of the line...Ch. 12.3 - Tangent line at the origin Find the polar equation...Ch. 12.3 - Prob. 22ECh. 12.3 - Multiple tangent lines at a point a. Give the...Ch. 12.3 - Multiple tangent lines at a point a. Give the...Ch. 12.3 - Horizontal and vertical tangents Find the points...Ch. 12.3 - Horizontal and vertical tangents Find the points...Ch. 12.3 - Horizontal and vertical tangents Find the points...Ch. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - Prob. 31ECh. 12.3 - Prob. 32ECh. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Intersection points and area a. Find all the...Ch. 12.3 - Intersection points and area a. Find all the...Ch. 12.3 - Intersection points and area a. Find all the...Ch. 12.3 - Intersection points and area a. Find all the...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Areas of regions Make a sketch of the region and...Ch. 12.3 - Area of plane regions Find the areas of the...Ch. 12.3 - Area of plane regions Find the areas of the...Ch. 12.3 - Area of plane regions Find the areas of the...Ch. 12.3 - Area of plane regions Find the areas of the...Ch. 12.3 - Area of polar regions Find the area of the regions...Ch. 12.3 - Area of polar regions Find the area of the regions...Ch. 12.3 - Prob. 59ECh. 12.3 - Area of polar regions Find the area of the regions...Ch. 12.3 - Two curves, three regions Determine the...Ch. 12.3 - Prob. 62ECh. 12.3 - Arc length of polar curves Find the length of the...Ch. 12.3 - Prob. 64ECh. 12.3 - Prob. 65ECh. 12.3 - Prob. 66ECh. 12.3 - Prob. 67ECh. 12.3 - Arc length of polar curves Find the length of the...Ch. 12.3 - Arc length of polar curves Find the length of the...Ch. 12.3 - Arc length of polar curves Find the length of the...Ch. 12.3 - Prob. 71ECh. 12.3 - Prob. 72ECh. 12.3 - Prob. 73ECh. 12.3 - Prob. 74ECh. 12.3 - Prob. 75ECh. 12.3 - Prob. 76ECh. 12.3 - Prob. 77ECh. 12.3 - Prob. 78ECh. 12.3 - Prob. 79ECh. 12.3 - Prob. 80ECh. 12.3 - Regions bounded by a spiral Let Rn be the region...Ch. 12.3 - Tangents and normals Let a polar curve be...Ch. 12.3 - Prob. 83ECh. 12.3 - Prob. 84ECh. 12.3 - Grazing goat problems Consider the following...Ch. 12.3 - Grazing goat problems Consider the following...Ch. 12.3 - Prob. 87ECh. 12.4 - Verify that x2+(yp)2=y+p is equivalent to x2 =...Ch. 12.4 - Prob. 2QCCh. 12.4 - In the case that the vertices and foci are on the...Ch. 12.4 - Prob. 4QCCh. 12.4 - Prob. 5QCCh. 12.4 - Prob. 6QCCh. 12.4 - Give the property that defines all parabolas.Ch. 12.4 - Prob. 2ECh. 12.4 - Give the property that defines all hyperbolas.Ch. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - What is the equation of the standard parabola with...Ch. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Given vertices (a, 0) and eccentricity e, what are...Ch. 12.4 - Prob. 10ECh. 12.4 - What are the equations of the asymptotes of a...Ch. 12.4 - Prob. 12ECh. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Prob. 16ECh. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Prob. 18ECh. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Prob. 27ECh. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Graphing conic sections Determine whether the...Ch. 12.4 - Prob. 31ECh. 12.4 - Equations of parabolas Find an equation of the...Ch. 12.4 - Equations of parabolas Find an equation of the...Ch. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Equations of parabolas Find an equation of the...Ch. 12.4 - From graphs to equations Write an equation of the...Ch. 12.4 - From graphs to equations Write an equation of the...Ch. 12.4 - Equations of ellipses Find an equation of the...Ch. 12.4 - Equations of ellipses Find an equation of the...Ch. 12.4 - Equations of hyperbolas Find an equation of the...Ch. 12.4 - Equations of hyperbolas Find an equation of the...Ch. 12.4 - Equations of ellipses Find an equation of the...Ch. 12.4 - Prob. 44ECh. 12.4 - Equations of hyperbolas Find an equation of the...Ch. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - From graphs to equations Write an equation of the...Ch. 12.4 - From graphs to equations Write an equation of the...Ch. 12.4 - Prob. 51ECh. 12.4 - Golden Gate Bridge Completed in 1937, San...Ch. 12.4 - Eccentricity-directrix approach Find an equation...Ch. 12.4 - Eccentricity-directrix approach Find an equation...Ch. 12.4 - Eccentricity-directrix approach Find an equation...Ch. 12.4 - Eccentricity-directrix approach Find an equation...Ch. 12.4 - Prob. 57ECh. 12.4 - Prob. 58ECh. 12.4 - Prob. 59ECh. 12.4 - Prob. 60ECh. 12.4 - Prob. 61ECh. 12.4 - Prob. 62ECh. 12.4 - Tracing hyperbolas and parabolas Graph the...Ch. 12.4 - Tracing hyperbolas and parabolas Graph the...Ch. 12.4 - Tracing hyperbolas and parabolas Graph the...Ch. 12.4 - Tracing hyperbolas and parabolas Graph the...Ch. 12.4 - Prob. 67ECh. 12.4 - Hyperbolas with a graphing utility Use a graphing...Ch. 12.4 - Tangent lines Find an equation of the tine tangent...Ch. 12.4 - Prob. 70ECh. 12.4 - Tangent lines Find an equation of the tine tangent...Ch. 12.4 - Tangent lines Find an equation of the tine tangent...Ch. 12.4 - Tangent lines for an ellipse Show that an equation...Ch. 12.4 - Prob. 74ECh. 12.4 - Prob. 75ECh. 12.4 - Prob. 76ECh. 12.4 - Another construction for a hyperbola Suppose two...Ch. 12.4 - The ellipse and the parabola Let R be the region...Ch. 12.4 - Volume of an ellipsoid Suppose that the ellipse...Ch. 12.4 - Area of a sector of a hyperbola Consider the...Ch. 12.4 - Volume of a hyperbolic cap Consider the region R...Ch. 12.4 - Prob. 82ECh. 12.4 - Prob. 83ECh. 12.4 - Prob. 84ECh. 12.4 - Prob. 85ECh. 12.4 - Prob. 86ECh. 12.4 - Prob. 87ECh. 12.4 - Prob. 88ECh. 12.4 - Shared asymptotes Suppose that two hyperbolas with...Ch. 12.4 - Focal chords A focal chord of a conic section is a...Ch. 12.4 - Focal chords A focal chord of a conic section is a...Ch. 12.4 - Focal chords A focal chord of a conic section is a...Ch. 12.4 - Prob. 93ECh. 12.4 - Prob. 94ECh. 12.4 - Confocal ellipse and hyperbola Show that an...Ch. 12.4 - Approach to asymptotes Show that the vertical...Ch. 12.4 - Prob. 97ECh. 12.4 - Prob. 98ECh. 12 - Explain why or why not Determine whether the...Ch. 12 - Prob. 2RECh. 12 - Prob. 3RECh. 12 - Eliminating the parameter Eliminate the parameter...Ch. 12 - Prob. 5RECh. 12 - Prob. 6RECh. 12 - Parametric curves and tangent lines a. Eliminate...Ch. 12 - Parametric curves and tangent lines a. Eliminate...Ch. 12 - Prob. 9RECh. 12 - Parametric curves a. Eliminate the parameter to...Ch. 12 - Parametric curves a. Eliminate the parameter to...Ch. 12 - Prob. 12RECh. 12 - Tangent lines Find an equation of the line tangent...Ch. 12 - Parametric descriptions Write parametric equations...Ch. 12 - Parametric description Write parametric equations...Ch. 12 - Parametric description Write parametric equations...Ch. 12 - Parametric description Write parametric equations...Ch. 12 - Parametric description Write parametric equations...Ch. 12 - Area bounded by parametric curves Find the area of...Ch. 12 - Area bounded by parametric curves Find the area of...Ch. 12 - Prob. 21RECh. 12 - Arc length Find the length of the following...Ch. 12 - Arc length Find the length of the following...Ch. 12 - Prob. 24RECh. 12 - Sets in polar coordinates Sketch the following...Ch. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Prob. 29RECh. 12 - Prob. 30RECh. 12 - Polar curves Graph the following equations. 31. r...Ch. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Polar conversion Write the equation...Ch. 12 - Polar conversion Consider the equation r = 4/(sin ...Ch. 12 - Prob. 37RECh. 12 - Prob. 38RECh. 12 - Prob. 39RECh. 12 - Slopes of tangent lines a. Find all points where...Ch. 12 - Slopes of tangent lines a. Find all points where...Ch. 12 - Prob. 42RECh. 12 - Prob. 43RECh. 12 - The region enclosed by all the leaves of the rose...Ch. 12 - Prob. 45RECh. 12 - The region inside the limaon r = 2 + cos and...Ch. 12 - Areas of regions Find the ares of the following...Ch. 12 - Prob. 48RECh. 12 - The area that is inside the cardioid r = 1 + cos ...Ch. 12 - Prob. 50RECh. 12 - Prob. 51RECh. 12 - Arc length of the polar curves Find the...Ch. 12 - Prob. 53RECh. 12 - Prob. 54RECh. 12 - Conic sections a. Determine whether the following...Ch. 12 - Prob. 56RECh. 12 - Prob. 57RECh. 12 - Tangent lines Find an equation of the line tangent...Ch. 12 - Tangent lines Find an equation of the line tangent...Ch. 12 - Prob. 60RECh. 12 - Prob. 61RECh. 12 - Prob. 62RECh. 12 - Prob. 63RECh. 12 - Prob. 64RECh. 12 - Prob. 65RECh. 12 - Prob. 66RECh. 12 - Eccentricity-directrix approach Find an equation...Ch. 12 - Prob. 68RECh. 12 - Prob. 69RECh. 12 - Prob. 70RECh. 12 - Prob. 71RECh. 12 - Prob. 72RECh. 12 - Prob. 73RECh. 12 - Prob. 74RECh. 12 - Lam curves The Lam curve described by...Ch. 12 - Prob. 76RE
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- True or False:- in the given question , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false The line y = (1 -3 √0.5)x divides the region under the curve f (x) = x(1 - x)on [0, 1] into two regions of equal area.arrow_forwarda. Set up an integral for the area of the surface generated byrevolving the given curve about the indicated axis.b. Graph the curve to see what it looks like. If you can, graphthe surface too.c. Use your utility’s integral evaluator to find the surface’s areanumerically. x1/2 + y1/2 = 3 from (4, 1) to (1, 4); x-axisarrow_forwardPICTURE FOR SUPPORT Use the stoke’s theorem to evaluate the anti derivative where F = and S is the surface of the solid bounded by the paraboloid Z= 1-x^2-y^2 and Z=0, C is the bounded of S with counter clockwise orientation .arrow_forward
- StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).arrow_forwardLet S be the solid of revolution obtained by revolving about the x-axis the bounded region R enclosed by the curve y=e−2x and the lines x=−1, x=1 and y=0. We compute the volume of S using the disk method. a) Let u be a real number in the interval −1≤x≤1. The section x=u of S is a disk. What is the radius and area of the disk? Radius: Area: b) The volume of S is given by the integral (b to a) ∫f(x)dx, where: a= b= and f(x)= c) Find the volume of S. Give your answer with an accuracy of four decimal places. Volume:arrow_forwardFine the area of the surface obtained by the rotating the following curve about the x-axis. (The interval is provided in the image.)arrow_forward
- Area bounded by parametric curves Find the area of the following regions. The region bounded by the y-axis and the parametric curvex = t2, y = t - t3, for 0 ≤ t ≤ 1arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cylinder y2 + z2 = 36, for 0 ≤ x ≤ 9arrow_forward6.Finding the Area of a Surface of Revolution. Set up and evaluate the definite integral forthe area of the surface generated by revolving the curve aboutthe y-axis. y=1−x²/4, 0≤x≤2 7.Use Implicit derivatives to find an equation of the tangent line to the curve sinx+cosy=1 at point (?/2,?/2). 5.Find the area of the region bounded by the graphs of x=3−y^2 and x=1+yarrow_forward
- Find a function f(x, y) such that f is continuous everywhere. the contour map of f consists of ellipses centered at the origin. (0,0, f(0, 0)) is the highest point on the surface z = f(x, y). Then find the range of f(x, y). Justify your answer using algebraic work.arrow_forwardSurface area using an explicit description Find the area ofthe following surfaces using an explicit description of the surface. The cone z2 = 4(x2 + y2), for 0 ≤ z ≤ 4arrow_forwardSurface integrals using an explicit description Evaluate the surface integral ∫∫S ƒ(x, y, z) dS using an explicit representation of the surface. ƒ(x, y, z) = 25 - x2 - y2; S is the hemisphere centered at theorigin with radius 5, for z ≥ 0.arrow_forward
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