Calculus Answers

A spring is such that it would be stretched 15.36 in by a 40 lb weight. Let the weight be attached

to a spring and pulled down 6.5 in below the equilibrium point. If the weight is started with an

upward velocity of 7 flt/sec, describe the motion. No damping force but an impressed force of

F(t) = 10lb is present.

A spring with constant 1.5lb/ft, lies on a long smooth (frictionless) table. An 8 lb weight is

attached to the spring and is at rest at equilibrium position. A 6 lb force is applied to the support

along the line of action of the spring for 5 secs and is removed. Discuss the motion.

A spring is such that it would be stretched 8 in by a 20 lb weight. Let the weight be attached to

a spring and pulled down 5 in below the equilibrium point. If the weight is started with an upward

velocity of 4 flt/sec, describe the motion. No damping force or impressed force is present.

Find by double integration the area of the region in 𝑥𝑦 plane bounded by the curves 𝑦 = 𝑥

2 and

𝑦 = 4𝑥 − 𝑥

2

.

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).

∬(𝑦 + 𝑥𝑦

−2)𝑑𝐴

𝑅

, 𝑅 = {(𝑥, 𝑦)|0 ≤ 𝑥 ≤ 2, 1 ≤ 𝑦 ≤ 2}

a) Define differentiation and integration in calculus. Also write down the

differences between them.

b) Write down some application of Calculus in CSE.

c) Describe geometrical meaning of definite integral with figure.

Find ∂ϕ

∂x

∂ϕ∂x where ϕ=x

2

+y

2

Show that the curve with parametric equations

x = sin t and y = sin(t + sin t) for 0 ≤ t ≤ 2π