1. Compute the volume of the solid formed by revolving the region bounded by y=2-x, x = 0, and y = 0: (a) about the x-axis. (b) about the line x = = 3.

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Later this semester, we will study the Heaviside "step" function H(t), which is defined to be S0 ift0 H(t) = (See the image above for the graph of the Heaviside function.) Find all possible solutions (if any) to the initial value problem below, where H(t) is the Heaviside function: y (t) = H(t)y(t), with y(0) = yo- In your work, carefully specify the number of solutions for each possible value of yo. Note: since H(t) is a discontinuous function, Theorems 2.4.1 and 2.4.2 in our textbook do not apply. So, in theory, anything could happen! Hints: since the values of H(t) are either 0 or 1, first find all possible solutions to y = 0y and separately to y = 1y. Then see if any two of those solutions can be "pieced together" to make a single solution y(t) to y = H(t) y with the given initial condition that y(0) = yo. Another Hint: a "solution" to an initial value problem must be differentiable on all of its domain.

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