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Changing the Order of
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Multivariable Calculus
- Current Attempt in Progress Locate the centroid of the shaded area. Set b = 0.30 a. b Answer: x=0(1-2²) a (x, y) = (i x ) aarrow_forwardMass of a box A solid box D is bounded by the planes x = 0, x = 3,y = 0, y = 2, z = 0, and z = 1. The density of the box decreases linearly in the positive z-direction and is given by ƒ(x, y, z) = 2 - z. Find the mass of the box.arrow_forwardproof that S a² + y) dA a (3a + 4) 36 Where is the region defined by the functions y = x, y 0, y= a, a>0arrow_forward
- Find the centroid of the region bounded by the graphs of the functions y = = 3x², y = x² +5 The centroid is at (x, y) where T= 0 65 y= 16 Question Help: Video Message instructor Xarrow_forwardDetermine the x- and y-coordinates of the centroid of the shaded area. y = 1+ -x - 1 2.arrow_forwardArea of Plane Region 3. R: x2 + 3y = 4 and x − 2y = 4.4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forward
- mtegrals ▸ 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_forwardTutorial Exercise Find the area of the surface. The portion of the cone z = 7 x² + y2 inside the cylinder x² + y2 = 9 Step 1 The definition of the surface area says if f and its first partial derivatives are continuous on the closed interval R in the xy-plane, then the area of the surface S given by z = f(x, y) over R is S = ds = / | V1+ [f,(x, y)1² + [f,cx, v)]? ds. We are asked to find the area of the portion of the cone z = 7VX2 + y² inside the cylinderx? + y2 = 9. Step 2 To findf, (x, y), partially differentiate f(x, y) with respect to x. x2 + y = дх 7x Vx² + y² Similarly, find f,(x, y). F,lx, y) = (7V; x² y2 ду 7 Therefore, 49x2 + 49 V1 + [f,(x, y)]² + [f,(x, y)]² 1 + x2 + y2 x² + y2 Simplifying, we have V1+ [f,(x, y)]² + [f,(x, v)]² =arrow_forwardSketch the reglon R of integration and switch the order of Integration. V 16 - x f(x, y) dy dx 2 -2 2 2 -D4 -2 V16-x2 f(x, y) dy dx = (x, Y) dx dy 16 - yarrow_forward
- (a) Sketch the region of integration R in the xy - plane and sketch the region G in the uv - plane using the coordinate transformation x = 2u and y = 2u + 4v.arrow_forwardTutorial Exercise Use the given transformation to evaluate the given integral, where R is the triangular region with vertices (0, 0), (6, 1), and (1, 6). (х — Зу) dA, х %3D би + v, у %3Dи + 6у Step 1 For the transformation x = 6u + v, y = u + 6v, the Jacobian is дх дх дх, у) — a(u, v) dv ду ду du = 35 du dv Also, х — Зу %3D (6и + v) — 3(u + бv) %3D Зи — V. Submit Skip (you cannot come back).arrow_forwardry dA where D is the triangular region with vertices (0,0), (1,0), (0,3) Evaluate the double integral I = Darrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,