(10 points) In this problem you will derive the formula for the Laplace transform of the second derivative of a function y. Use y and y′ for y(t) and y′(t), y0 and y1 for the initial conditions y(0) and y′(0), and Y for the Laplace transform of y. L{y′′(t)}(s)=∫[infinity]0 e−t y′′(t)dt

By definition of the Laplace transform,

Integrate by parts:

Integrate by parts again to compute the transform of :

Putting everything together, we have

or in terms of the given initial conditions,

the volume is the product of the area of the base, 3x(3x+4), and the height, 3x-1

step-by-step explanation: