Question

12. The temperature at any point on a metal plate in the xy plane is given by T(x, y) = 100 – 4x2 - y2, where x and y are measured in inches and T in degrees Celsius. Consider the portion of the plate that lies on the rectangular region R = (1,5] x [3, 6]. Write an iterated integral whose value represents the volume under the surface T over the rectangle R. b. Evaluate the iterated integral you determined in (a). Find the area of the rectangle, R. d. Determine the exact average temperature, TAVG(R), over the region R. a. C.

Parts c and d only please

Answer 1

> ② Given , Given, (^,y) = 100- 4x² - y² R= [15] *[3,6]. The volume under the rurface T over the rectangle R is given by, SST(x,y) dady VE R 5 6 => Va 1 (100-42=ye) dy dx 3 - 5 6 6 Now, V = 5 6 5 5 6 6 TOO dal dy Scoo 4x² - y²) dydze a betyg 00[] [4] 3 [13] [%] 100(5-1)(5-3) - 5 (125-1)(6-3) - $(5-1)(216 – 27) 3 I 3 3 - de fyddy - [*]*[y], 6 100 3 1200 496 252 = 452 © Asea of the rectangle R, A=SS andy R 5 6 3 3 (dej dy [] [] = (5-1) (6-3) [12 temperature, T. 121 over R, a The exact average is given by, v 113 3 = 452 12 Tava(P) A