Finding u and du/dx In Exercises 1-8, identify u and
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Calculus: An Applied Approach (MindTap Course List)
- 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_forwardDouble 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_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) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forward
- Scalar line integrals Evaluate the following line integral along the curve C.arrow_forwardUsing polar coordinates, evaluate the integral (sin(x2+y2)dA) over the region 1<=x2+y2<=81.arrow_forwardUsing Cauchy's Theorem calculate the following integral and the singular points of the function, where C: z(t) = 3*cost(t) + i*(3+ 3*sin(t)) 0 < t < 2πarrow_forward
- Evaluating 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, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_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, z) = (x2 + y2 + z2)/2; C: r(t) = ⟨cos t, sin t, t/π⟩ , for 0 ≤ t ≤ 2πarrow_forwardSetup an integral to find the surface area for the graph y = x1/2 rotated about the y axis under the restriction that 1 < x < 4.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning