To evaluate the following integral, carry out these steps.

a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables.
b. Find the limits of integration for the new integral with respect to u and v.
c. Compute the Jacobian.
d. Change variables and evaluate the new integral.

Modifying Below Integral from nothing to nothing Integral from nothing to nothing With Upper R∫∫Rx squared StartRoot x plus 2y EndRoot 2x+2y dA, where Upper R equals StartSet (x comma y ): 0 less than or equals x less than or equals 2 comma negative StartFraction x Over 2 EndFraction less than or equals y less than or equals 1 minus x EndSetR=(x, y): 0≤x≤2, − x 2≤y≤1−x; use x equals 2 ux=2u, y equals v minus uy=v−u.

Amachine purchased three years ago for $720,000 has a current book value using straight-line depreciation of $400,000: its operating expenses are $60,000 per year. a replacement machine would cost $480,000, have a useful life of nine years, and would require $26,000 per year in operating expenses. it has an expected salvage value of $130,000 after nine years. the current disposal value of the old machine is $170,000: if it is kept 9 more years, its residual value would be $20,000. calculate the total costs in keeping the old machine and purchase a new machine. should the old machine be replaced?

Burke enterprises is considering a machine costing $30 billion that will result in initial after-tax cash savings of $3.7 billion at the end of the first year, and these savings will grow at a rate of 2 percent per year for 11 years. after 11 years, the company can sell the parts for $5 billion. burke has a target debt/equity ratio of 1.2, a beta of 1.79. you estimate that the return on the market is 7.5% and t-bills are currently yielding 2.5%. burke has two issuances of bonds outstanding. the first has 200,000 bonds trading at 98% of par, with coupons of 5%, face of $1000, and maturity of 5 years. the second has 500,000 bonds trading at par, with coupons of 7.5%, face of $1000, and maturity of 12 years. kate, the ceo, usually applies an adjustment factor to the discount rate of +2 for such highly innovative projects. should the company take on the project?

Suppose an economy has only two sectors: goods and services. each year, goods sells 80% of its outputs to services and keeps the rest, while services sells 62% of its output to goods and retains the rest. find equilibrium prices for the annual outputs of the goods and services sectors that make each sector's income match its expenditures.