Minimizing a Double
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Multivariable Calculus
- YOUR TURN Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by y=x2+1, y=0, x=1 and x=1.arrow_forwardA soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.arrow_forwardFind the area of the region bounded by x = y³ - 4y² + 3y and the x = y²-y. (a) A square units 77 12 12 (b) A= square units (c) A= 71 6 square units (d) A= square unitsarrow_forward
- Maximizing a double integral What region R in the xy-plane maximizes the value of /| (4 – x² – 2y²) dA? Give reasons for your answer.arrow_forwardmtegrals ▸ Example 4 Evaluate ff.(2x. (2x - y²) dA R over the triangular region R enclosed between the lines y = -x + 1, y = x + 1, and y = 3. dx dy izontal line correspondingarrow_forwardSet-up the integral for the area of the plane region bounded by y=x+4 and y=x²-2x- (x²-3x - 4)dx (-x²+3x+4)dx -1 8 (5+ √y+1-y)dy -1 8 (5+√y+1-y)dy -1 3arrow_forward
- Area of Plane Region 3. R: x2 + 3y = 4 and x − 2y = 4.4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forwardEvaluae the double intergal for the function f(x,y) and the given region R f(x,y)=7xe^-y^2; R is bounded by x=0, x= √y, and y=2arrow_forwardSketch the region bounded by the graphs of the equations (include representative rectangle) f (x) = 3, g(x) = x – 1,and find the area of the region. Vx -1arrow_forward
- y (1, 1) X=1 y=x² C X (0,0) Region A is bounded by the parabola x = y², the line y = 1, and the y-axis. Region B is bounded by the parabolas x = y² and y = x². Region C is bounded by the parabola y = x², the line x = 1, and the x-axis. 1. What is the area of A? A. squnit B. 2 squnit C. squnit D. 3 squnit 2. What is the area of B? A. squnit B. 3 squnit C. squnit D. squnit 3. What is the area of C? A. 2 squnit B. squnit C. squnit D. / squnit 4. What is the volume of the solid generated when region C is revolved about the x-axis? A. V≈ 0.63 c. u. B. V≈ 0.72 c. u. D. V≈ 0.92 c. u. C. V≈ 0.84 c. u. 5. What is the volume of the solid generated when region B is revolved about the y-axis? A. V 2.13 c. u. B. V≈ 0.94 c. u. C.V ~ 1.27 c. u. D. V 3.08 c. u. y = 1 x = y² Barrow_forwardRis the region bounded by the functions f(x) = x² + 1 and g(x) = Væ + 1 and the lines a = 0 and a 3. Represent the volume when Ris rotated around the x-axis. 3 Volume = dx Use pi for "T" and sqrt(x) for "Va" and (x+1)^2 for "(x + 1)²"arrow_forwardUse a change of variables to find double integral R = sqrt((x-y)/(x+y+1)) dA, where R is the square with vertices (0, 0), (1, − 1), (2, 0), and (1, 1). u=x-y v=x+y Graph the region in xy-plane Graph the region in uv-plane Find the Jacobian State the converted integralarrow_forward
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