Approximation In Exercises 3-6, approximate the
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Multivariable Calculus
- Double integral to line integral Use the flux form of Green’sTheorem to evaluate ∫∫R (2xy + 4y3) dA, where R is the trianglewith vertices (0, 0), (1, 0), and (0, 1).arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The wedge in the first octant bounded by the cylinder x = z2 andthe planes z = 2 - x, y = 2, y = 0, and z = 0arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The solid between the sphere x2 + y2 + z2 = 19 and the hyperboloidz2 - x2 - y2 = 1, for z > 0arrow_forward
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The prism in the first octant bounded by z = 2 - 4x and y = 8.arrow_forwardEngineering Mechanics - Centroids Using Centroid by Integration, determine the x- and y-coordinates of the centroid of the shaded area.arrow_forward*INTEGRAL CALCULUS Show complete solution (with graph). 2. Determine the centroid of the area bounded by x^2 − y = 0 and x − y = 0.3. Determine the centroid of the area bounded by 2(y^2 + 4) − 2x − 8 = 0 and 8y + x^2 = 0.arrow_forward
- Channel flow The flow in a long shallow channel is modeled by the velocity field F = ⟨0, 1 - x2⟩, where R = {(x, y): | x | ≤ 1 and | y | < 5}.a. Sketch R and several streamlines of F.b. Evaluate the curl of F on the lines x = 0, x = 1/4, x = 1/2, and x = 1.c. Compute the circulation on the boundary of the region R.d. How do you explain the fact that the curl of F is nonzero atpoints of R, but the circulation is zero?arrow_forwardSix orderings Let D be the solid in the first octant bounded bythe planes y = 0, z = 0, and y = x, and the cylinder 4x2 + z2 = 4.Write the triple integral of ƒ(x, y, z) over D in the given order of integration. dy dz dxarrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by x = 0, x = 2, y = 0, y = e-z, z = 0, and z = 1arrow_forward
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by x = 0, x = 2, y = z, y = z + 1, z = 0, and z = 4arrow_forwardA. State the F undamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations x = t, y = t, z = 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forward*INTEGRAL CALCULUS Show complete solution (with graph) 8. Determine the centroid, C(x̅, y̅, z̅), of the solid formed in the first octant bounded by z + y − 16 = 0 and 2x^2 − 2(16 − y) =0.arrow_forward
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