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Calculus: Early Transcendental Functions
- Double integrate under z=xy, above the triangle with vertices (0,1),(0,4),(1,1).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_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
- 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_forwardFind 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_forward2 2 2),-0 Exercises: Evaluate and Sketch the region of integration and write an equivalent double of integration reversed. 14-2x 1. dydx 0 2 2. [| dxdy 11-x 3. dyck 0 1-x 4. dydx 2 2x 5-| (4x+2)dydxarrow_forward
- Sketch 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_forwardArea of Plane Region 2. R: y = 6x − x2and y = x2 − 2x.3. R: x2 + 3y = 4 and x − 2y = 4.4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forwarduv Use the map G(u, v) = to compute (x + y) dx dy, where D is the region bounded by x + y = 3, x + y = 6, y = x, y = 7x. v + 1 v + 1arrow_forward
- Sketch the region R of integration and switch the order of integration. f(x, у) dy dx 5- y 1- 3- y 2- -1 -1- -3 -2 -1 1 3 -2 -1- 4. 4- 3 3- y y 2- 3. -3 -2 1 2. 3 X. -12arrow_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_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_forward
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