Consider a wooden block in the shape of a cube whose edges are 10 cm long. The density of the wood is 0.8 g/cm3. The block is submersed in water; a guiding mechanism guarantees that the top and the bottom surfaces of the block are parallel to the surface of the water at all times. Let jc(/) be the depth of the block in the water at time t. Assume that jc is between 0 and 10 at all times.4')\a. Two forces are acting on the block: its weight and the buoyancy (the weight of the displaced water). Recall that the density of water is 1 g/cm3. Find formulas for these two forces. b. Set up a differential equation for x (/). Find the solution, assuming that the block is initially completely submersed ( j c ( 0 ) = 1 0 ) and at rest. c. How does the period of the oscillation change if you change the dimensions of the block? (Consider a larger or smaller cube.) What if the wood has a different density or if the initial state is different? What if you conduct the experiment on the moon?

Calculus III Notes Week 8 Post Midterm, Chapter 15 Section 1 and 2 15.1 Double Integral Introduction -Definite Integral Review For a single variable function f, the integral from a to b can be looked at as a Riemann sum. Dividing the interval a,b into subintervals of equal width and taking sample values of f at these various subintervals allows us to calculate the integral with the formula ∑ f ( x ) Δ x If f is positive along the interval a,b, the Riemann sum can be looked at as the sum of the areas of many rectangles with equal width Δ x and height f (x). -Solving for Volume Using Double Integrals For a function f of two variables, the volume within a closed area can be found by a similar method to the Riemann