Evaluating a Line
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Calculus (MindTap Course List)
- Calculus In Exercises 65-68, show that f and g are orthogonal in the inner product space C[a,b]with the inner product f,g=abf(x)g(x)dx. C[/2,/2], f(x)=cosx, g(x)=sinxarrow_forwardEvaluate F · dr using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. (2z + 4y) dx + (4x – 3z) dy + (2x – 3y) dz (a) C: line segment from (0, 0, 0) to (1, 1, 1) (b) C: line segment from (0, 0, 0) to (0, 0, 1) to (1, 1, 1) (c) C: line segment from (0, 0, 0) to (1, 0, 0) to (1, 1, 0) to (1, 1, 1)arrow_forwardCalculate the line integral of the vector field F = (y, x,x² + y² ) around the boundary curve, the curl of the vector field, and the surface integral of the curl of the vector field. The surface S is the upper hemisphere x² + y + z? = 25, z 2 0 oriented with an upward-pointing normal. (Use symbolic notation and fractions where needed.) F. dr = curl(F) =arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5 y dx + 5 x²dy, where Cis the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) oriented counterclockwise. + iarrow_forwardEvaluating Polar Integrals In Exercises 9-22, change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. μl pV²-3² 11 12. Jo Jo ra I √a²-x² тугилау dy dx JOJOarrow_forward人工知能を使用せず、 すべてを段階的にデジタル形式で解決してください。 ありがとう SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE CHATGPT Find the integral of the vector function F(t)=(f.,cost)arrow_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) = (x2 + y2 + z2)/2; C: r(t) = ⟨cos t, sin t, t/π⟩ , for 0 ≤ t ≤ 2πarrow_forwardulus III |Uni Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively oriented curve C, where C is the rectangle with vertices(0,0), (4, 0), (4, 2) and (0, 2).arrow_forwardDifferentiation of Vector-Valued Functions In Exercises 7 and 8, find r (t), r(t,), and r (t,) for the given value of t. Then sketch the space curve represented by the vector-valued function, and sketch the vectors r(t) and r'(t). 7. r(t) = 2 cos ti + 2 sin tj + tk, toarrow_forward
- Evaluating line integrals Evaluate the line integral ∫C F ⋅ drfor the following vector fields F and curves C in two ways.a. By parameterizing Cb. By using the Fundamental Theorem for line integrals, if possible F = ∇(xyz); C: r(t) = ⟨cos t, sin t, t/π⟩ , for 0 ≤ t ≤ πarrow_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_forwardOutward flux of a radial field Use Green’s Theorem to compute the outward flux of the radial field F = ⟨x, y⟩ across the unit circle C = {(x, y2: x2 + y2 = 1} (see figure). Interpret the result.arrow_forward
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