Evaluate the line integral, where C is the given curve.   xyeyz dy, C: x = 4t, y = 3t2, z = 3t3, 0 ≤ t ≤ 1 C?

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  • 4 weeks ago

    ∫c xy e^(yz) dy

    = ∫ (4t)(3t²) e^[(3t²)(3t³] 6t dt ……… bounds t = 0 to t = 1

    = ∫ 72t⁴ e^(9t⁵) dt

    Let u = 9t⁵

    du = 45t⁴ dt

    and, trying to show the transition steps, the integral becomes

    = ∫ (1/45) * 72t⁴ e^(9t⁵) * 45 dt

    = ∫ (72/45) e^(9t⁵) 45t⁴ dt

    = ∫ (72/45) e^u du

    = (72/45) * [e^u]

    = (72/45) * [e^(9t⁵)] …….. evaluated at t = 0 and t = 1

    = (72/45) * [(e^9) - 1]

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