ha (lvh262) – Homework 15.1 – karakurt – (56295) 1This print-out should have 3 questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.001 10.0 pointsEvaluate the double integralI =Z ZA6 dxdywithA =n(x, y) : 2 ≤ x ≤ 6, 2 ≤ y ≤ 5oby first identifying it as the volume of a solid.1. I = 762. I = 743. I = 784. I = 72 correct5. I = 80Explanation:The value of I is the volume of the solidbelow the graph of z = f (x, y) = 6 and abovethe regionA =n(x, y) : 2 ≤ x ≤ 6, 2 ≤ y ≤ 5o.Since A is a rectangle, this solid is a box w ithbase A and height 6 . Its volume, therefore, isgiven bylength × width × height= (6 − 2) × (5 − 2) × 6 .Consequently,I = 72 .keywords: volume, double integral, rectangu-lar region, rectangular solid002 10.0 pointsThe graph of the functionz = f (x, y) = 3 − xis the plane shown inz3xyDetermine the value of the double integralI =Z ZAf(x, y) dxdyover the regionA =n(x, y) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 2oin the xy-pla ne by first identifying it as thevolume of a solid below the graph of f .1. I = 11 cu. units2. I = 10 cu. units3. I = 12 cu. units4. I = 9 cu. units correct5. I = 13 cu. unitsExplanation:The double integralI =Z ZAf(x, y) dxdyis the volume of the soli d below the graph off having the rectangleA =n(x, y) : 0 ≤ x ≤ 3, 0 ≤ y ≤ 2oha (lvh262) – Homework 15.1 – karakurt – ( 56295) 2for its base. Thus the solid is the wedgez33xy(3, 2)and so its volume is the area of triangularface multiplied by the thickness of the wedge.Consequently,I = 9 cu. units .keywords: double integral, linear function,volume under graph, volume, r ecta ngular re-gion, prism, triangle003 10.0 pointsDetermine the value of the double integralI =Z ZR(6 − x) dxdyover the regionR = {(x, y) : 4 ≤ x ≤ 6 , 0 ≤ y ≤ 2}in the xy-plane by first identifying it as thevolume of a solid.1. I = 32. I = 63. I = 4 correct4. I = 55. I = 2Explanation:The double integral I is the volume of thesolid below the graph ofz = 6 − xhaving the rectangleR = {(x, y) : 4 ≤ x ≤ 6 , 0 ≤ y ≤ 2}as its base. Thus the solid is the wedgez62xy(6, 2)having a triangular face of height 2 and base2. Since the wedge has length 2, the solid thushasvolume = 4
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