Solve the initial value problem.Differential Equation:  = (sec2t)i + (4t3- 3)jInitial Condition: r(0) = -6j

Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.The curve y = ln x and the lines x = 8 and y = 0

Find the volume by using polar coordinates.The region bounded by the paraboloid z = x2 + y2, the cylinder x2 + y2 = 49, and the xy-plane.

Find the volume of the indicated region.the region that lies under the plane z = 4x + 9y and over the triangle with vertices at (1, 1), (2, 1) and (1, 2) Give an EXACT answer, no decimal approximations. If necessary, use the forward slash, /, for fractions.

Find the average value of the function f over the given region.; R = {(x, y): 1 ≤ x ≤ 4, 1 ≤ y ≤ 4}

Find the average value of the function f over the given region.; R = {(x, y): 1 ≤ x ≤ 5, 1 ≤ y ≤ 5}

Find the average value of the function f over the given region.; R = {(x, y): 1 ≤ x ≤ 3, 1 ≤ y ≤ 3}

Solve the problem.Integrate

Find the volume by using polar coordinates.The region bounded by the paraboloid z = x2 + y2 ,  the cylinder x2 + y2 = 64, and the xy-plane.

Solve the problem.Let R ={(x, y): 1 ≤ x < 5, 0 ≤ y < 6}. Evaluate