Problem 1QC: Explain mathematically why differentiating the arc length integral leads to s'(t) = |r'(t)|. Problem 2QC: Suppose r(t) = t,0, for a t b, is a parametric description of C; note that C is the interval [a,... Problem 3QC Problem 4QC: Suppose a two dimensional force field is everywhere directed outward from the origin, and C is a... Problem 5QC Problem 1E: How does a line integral differ from the single-variable integral abf(x)dx? Problem 2E: If a curve C is given by r(t) = t, t2, what is |r(t)|? Problem 3E Problem 4E: Find a parametric description r(t) for the following curves. 4. The segment of the curve x = sin y... Problem 5E: Find a parametric description r(t) for the following curves. 5. The line segment from (1, 2, 3) to... Problem 6E: Find a parametric description r(t) for the following curves. 6. The quarter-circle from (1,0) to... Problem 7E: Find a parametric description r(t) for the following curves. 7. The segment of the parabola x = y2 +... Problem 8E: Find an expression for the vector field F = x y, y x (in terms of t) along the unit circle r(t) =... Problem 9E: Suppose C is the curve r(t) = t,t3, for 0 t 2, and F = x,2y. Evaluate CFT ds using the following... Problem 10E: Suppose C is the circle r(l) = cos l, sin l , for 0 t 2, and F = 1, x. Evaluate C F n ds Using the... Problem 11E: State two other forms for the line integral CFTds given that F = f, g, h. Problem 12E: Assume F is continuous on a region containing the smooth curve C from point A to point B and suppose... Problem 13E: Assume F is continuous on a region containing the smooth curve C from point A to point B and suppose... Problem 15E: How is the circulation of a vector field on a closed smooth oriented curve calculated? Problem 16E Problem 17E: Scalar line integrals Evaluate the following line integrals along the curve C. Cxy ds; C is the unit... Problem 18E: Scalar line integrals Evaluate the following line integrals along the curve C. C(x22y2) ds; C is the... Problem 19E: Scalar line integrals Evaluate the following line integrals along the curve C. 19. C(2x+y)ds; C is... Problem 20E: Scalar line integrals Evaluate the following line integrals along the curve C. 20. Cx ds; C is the... Problem 21E: Scalar line integrals Evaluate the following line integrals along the curve C. 21. Cxy3ds; C is the... Problem 22E: Scalar line integrals Evaluate the following line integrals along the curve C. 22. C3x cos y ds; C... Problem 23E: Scalar line integrals Evaluate the following line integrals along the curve C. 23. C(yz)ds; C is the... Problem 24E: Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the... Problem 25E: Scalar line integrals Evaluate the following line integrals along the curve C. 25. C(x2+y2)ds; C is... Problem 26E: Scalar line integrals Evaluate the following line integrals along the curve C. 26. C(x2+y2)ds; C is... Problem 27E: Scalar line integrals Evaluate the following line integrals along the curve C. 27. Cxx2+y2ds; C is... Problem 28E: Scalar line integrals Evaluate the following line integrals along the curve C. 28. C(xy)1/3ds; C is... Problem 29E: Scalar line integrals Evaluate the following line integrals along the curve C. 29. Cxy ds; C is a... Problem 30E: Scalar line integrals Evaluate the following line integrals along the curve C. 30. C(2x3y) ds; C is... Problem 31E: Scalar line integrals Evaluate the following line integrals along the curve C. 31. C(x+y+z)ds; C is... Problem 32E: Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the... Problem 33E: Scalar line integrals Evaluate the following line integrals along the curve C. 33. Cxzds; C is the... Problem 34E: Scalar line integrals Evaluate the following line integrals along the curve C. 34. Cxeyzds; C is... Problem 35E: Mass and density A thin wire represented by the smooth curve C with a density (units of mass per... Problem 36E: Mass and density A thin wire represented by the smooth curve C with a density (units of mass per... Problem 37E: Average values Find the average value of the following functions on the given curves. 21.f(x,y)=x+2y... Problem 38E: Average values Find the average value of the following functions on the given curves. 24.f(x,y)=xey... Problem 39E: Length of curves Use a scalar line integral to find the length of the following curves.... Problem 40E: Length of curves Use a scalar line integral to find the length of the following curves.... Problem 41E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 42E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 43E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 44E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 45E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 46E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 47E: 47–48. Line integrals from graphs Determine whether along the paths C1 and C2 shown in the... Problem 48E: Line integrals from graphs Determine whether c F dr along the paths C1 and C2 shown in the... Problem 49E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 50E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 51E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 52E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 53E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 54E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 55E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 56E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 57E: Circulation Consider the following vector fields F and closed oriented curves C in the plane (see... Problem 58E: Circulation Consider the following vector fields F and closed oriented curves C in the plane (see... Problem 59E: Flux Consider the vector fields and curves in Exercises 5758. a. Based on the picture, make a... Problem 60E: Flux Consider the vector fields and curves in Exercises 5758. a. Based on the picture, make a... Problem 61E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 62E: Flying into a headwind An airplane flies in the xz-plane, where x increases in the eastward... Problem 63E: Flying into a headwind
How does the result of Exercise 62 change if the force due to the wind is F =... Problem 64E Problem 65E: Changing orientation Let f(x, y) = x and let C be the segment of the parabola y = x2 joining O(0, 0)... Problem 66E: Work in a rotation field Consider the rotation field F = y, x and the three paths shown in the... Problem 67E: Work in a hyperbolic field Consider the hyperbolic force field F = ⟨y, x⟩ (the streamlines are... Problem 68E: Assorted line integrals Evaluate each line integral using the given curve C. 68. Cx2dx+dy+ydz; C is... Problem 69E: Assorted line integrals Evaluate each line integral using the given curve C. 69.... Problem 71E: Assorted line integrals Evaluate each line integral using the given curve C. 71. Cyx2+y2dxxx2+y2dy;... Problem 72E: Assorted line integrals Evaluate each line integral using given curve C. 72. C(x+y)dx+(xy)dy+xdz;C... Problem 73E: Flux across curves in a vector field Consider the vector Field F = y, x shown in the figure.... Problem 74E Problem 75E: Zero circulation fields 57.Consider the vector field F = ax + by, cx + dy. Show that F has zero... Problem 76E: Zero flux fields 58.For what values of a and d does the vector field F = ax, dy have zero flux... Problem 77E: Zero flux fields 59.Consider the vector field F = ax + by, cx + dy. Show that F has zero flux across... Problem 78E: Heat flux in a plate A square plate R = {(x, y): 0 x 1, 0 y 1} has a temperature distribution... Problem 79E Problem 80E: Line integrals with respect to dx and dy Given a vector field F = ⟨f, 0⟩ and curve C with... Problem 81E: Looking ahead: Area from line integrals The area of a region R in the plane, whose boundary is the... Problem 82E format_list_bulleted