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  

Evaluate the double integral over the given region.

Which is NOT a main argument of the article?

According to Ojha, we should not let Barbie’s flaws deter us from finding her constructive lessons.