Finding u and du In Exercises 1–4, complete the table by identifying u and du for the
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Calculus
- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,xeax}arrow_forwardShowing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,ebx}, abarrow_forwardEvaluating line integrals using level curves Suppose the vector field F, whose potential function is φ, is continuous on ℝ2. Use the curves C1 and C2 and level curves of φ (see figure) to evaluate the following line integral.arrow_forward
- 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_forwardConverting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forwardcalc 3 Evaluate the integral below, where E lies between the spheres x2 + y2 + z2 = 16 and x2 + y2 + z2 = 25 in the first octant.arrow_forward
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- Line integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}arrow_forwardR zone being a square with vertices (0,2), (1,1), (2,2) and (1,3);Calculate the integral of the picture using the transformation u=x-y, v=x+y.arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = xy; C: r(t) = ⟨cos t, sin t⟩ , for 0 ≤ t ≤ πarrow_forward
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