Problem 1E: How does a line integral differ from the single-variable integral abf(x)dx? Problem 2E Problem 3E: If a curve C is given by r(t) = t, t2, what is |r(t)|? Problem 4E Problem 5E Problem 6E Problem 7E Problem 8E Problem 9E Problem 10E Problem 11E: Scalar line integrals with arc length as parameter Evaluate the following line integrals. 11.Cxyds;C... Problem 12E Problem 13E: Scalar line integrals with arc length as parameter Evaluate the following line integrals.... Problem 14E Problem 15E: Scalar line integrals in the plane a.Find a parametric description for C in the form r(t)=x(t),y(t).... Problem 16E Problem 17E: Scalar line integrals in the plane a.Find a parametric description for C in the form r(t)=x(t),y(t).... Problem 18E: Scalar line integrals in the plane a.Find a parametric description for C in the form r(t)=x(t),y(t).... Problem 19E: Scalar line integrals in the plane a.Find a parametric description for C in the form r(t)=x(t),y(t).... Problem 20E Problem 21E: Average values Find the average value of the following functions on the given curves. 21.f(x,y)=x+2y... Problem 22E: Average values Find the average value of the following functions on the given curves.... Problem 23E Problem 24E: Average values Find the average value of the following functions on the given curves. 24.f(x,y)=xey... Problem 25E: Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the... Problem 26E: Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the... Problem 27E Problem 28E: Scalar line integrals in 3Convert the line integral to an ordinary integral with respect to the... Problem 29E Problem 30E Problem 31E: Length of curves Use a scalar line integral to find the length of the following curves.... Problem 32E: Length of curves Use a scalar line integral to find the length of the following curves.... Problem 33E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 34E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 35E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 36E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 37E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 38E: Line integrals of vector fields in the plane Given the following vector fields and oriented curves... Problem 39E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 40E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 41E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 42E: Work integrals Given the force field F, find the work required to move an object on the given... Problem 43E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 44E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 45E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 46E: Work integrals in 3 Given the force field F, find the work required to move an object on the given... Problem 47E: Circulation Consider the following vector fields F and closed oriented curves C in the plane (see... Problem 48E Problem 49E: Flux Consider the vector fields and curves in Exercises 4748. a. Based on the picture, make a... Problem 50E Problem 51E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 52E: Flying into a headwind An airplane flies in the xz-plane, where x increases in the eastward... Problem 53E Problem 54E Problem 55E: Changing orientation Let f(x, y) = x and let C be the segment of the parabola y = x2 joining O(0, 0)... Problem 56E Problem 57E: Zero circulation fields 57.Consider the vector field F = ax + by, cx + dy. Show that F has zero... Problem 58E: Zero flux fields 58.For what values of a and d does the vector field F = ax, dy have zero flux... Problem 59E: Zero flux fields 59.Consider the vector field F = ax + by, cx + dy. Show that F has zero flux across... Problem 60E: Work in a rotation field Consider the rotation field F = y, x and the three paths shown in the... Problem 61E Problem 62E: Mass and density A thin wire represented by the smooth curve C with a density (units of mass per... Problem 63E: Mass and density A thin wire represented by the smooth curve C with a density (units of mass per... Problem 64E: Heat flux in a plate A square plate R = {(x, y): 0 x 1, 0 y 1} has a temperature distribution... Problem 65E Problem 66E: Flux across curves in a vector field Consider the vector Field F = y, x shown in the figure.... Problem 67E: Looking ahead: Area from line integrals The area of a region R in the plane, whose boundary is the... Problem 68E: Looking ahead: Area from line integrals The area of a region R in the plane, whose boundary is the... format_list_bulleted