Set, Up, but do not evaluate, an integral that represents the length of the foll

Question

Set, Up, but do not evaluate, an integral that represents the length of the following curve on the given interval. f(x) = Squareroot 4 + 2x^2 l lessthanorequalto x lessthanorequalto 8. Set up, but do not evaluate, integrals that represent the moment about the x-axis, M_x and the moment about the y-axis. M_y, of a thin plate of constant density partial differential covering the region bounded by the graphs: y = Squareroot x, y = 2, x = 1, and x = 4. A tank in the shape of a right circular cone is 14 feet tall and has a top radius of 4 feet. Suppose the cone is completely filled with water weighing 62.5 lb/ft^3. Set up, but do not evaluate, a definite integral that represents the work required to pump all but 2 feet of water over the edge of the tank. If a force of 80 N stretches a spring 2 meters beyond its natural length, how much work does it take to stretch the spring 5 meters beyond its natural length?

Explanation / Answer

# It is mentioned clearly that I don't need to evaluate the questions for final answer.

# There are 4 questions I am solving two out of them.

4)

We know that the length of a curve is given by:

L = a to b [1 + (dy/dx)²] dx

We are given that the curve is : y = (4 + 2x²)
So dy/dx = ½ * 1/(4 + 2x²) * 4x
or dy/dx = 2x /(4 + 2x²)
Now (dy/dx)2 = 4x2 / (4 + 2x²)

So now L = 1 to 8 [1 + 4x2 / (4 + 2x²)]dx

Now evaluate the integral for x = 1 to 8 to get curve length.

7)

By Hooke's law we know that the force needed to compress or extend a spring is directly proportional to the distance we stretch it.
Hooke's law can be represented as F = kx, where F is the force we apply and k is the spring constant (as per material) and x is the extension of the spring.

So in our question F = kx, gives
80 = k * 2
or k = 40 N/m

Now to find work done in stretching the spring to 5 m beyond its natural length, we will do integration like below :
Work dx

= F dx

= kx dx from x = 0 to 5

or Work = 40 * x dx [x = 0 to 5]

Now solve it to get answer.

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