Calculus Answers

.Find the average value of 𝑓(𝑥, 𝑦) = 𝑥

2𝑦 over the region 𝑅 which is a rectangle with vertices

(−1, 0), (−1, 5), (1, 5), (1, 0).

find inverse laplace transorm : se^-2s/(s^2 + pi^2)

∫ ∫ 𝑥𝑦𝑑𝑥𝑑𝑦 2𝑦

𝑦

2

find laplace transform : sin wt (0 < t< pi/w)

Find an approximate value of the double integral below where 𝑅 is the rectangular region having

vertices (−1, 1) and (2, 3). Take a partition of 𝑅 formed by the lines 𝑥 = 0, 𝑥 = 1, and 𝑦 = 2, and take (𝑢𝑖

, 𝑣𝑖) at the

center of the 𝑖th sub region.

∬(3𝑦 − 2𝑥

2)𝑑𝐴

𝑅

find laplace transform : 4u(t-pi) cost

fine laplace transform : e^-2t u(t-3)

fine laplace transforms of : (t-1) u(t-1)

[SADT8] If A

 and B

 are vector fields, prove the following:

∇(A⋅B)=(B⋅∇)A+(A⋅∇)B+B×(∇×A)+A×(∇×B).

A rectangular box whose volume is 32 is open at the top. If the surface of the area is 2( L + B )H +LB, where L L, B, H are length, breath and height respectively.

(A) Find the dimension of the box that may regure least material

(b) Investigate weather the dimension found require least material.