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find the surface area generated by rotating the given curve about the y-axis. x = et − t, y = 4et/2, 0 ≤ t ≤ 10

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Final answer:

To find the surface area of a shape generated by rotating a curve around the y-axis, set up an integral of 2π times x(t) times y(t) over the interval from 0 to 10 and evaluate it.

Step-by-step explanation:

Finding the Surface Area Generated by Rotating a Curve about the Y-axis

The question asks to find the surface area generated by rotating the curve given by the equations x = et - t and y = 4et/2, for 0 ≤ t ≤ 10, around the y-axis. To solve this problem, we use the formula for surface area of a shape generated by rotating a curve about the y-axis, which can be represented as an integral given by the formula:

S = 2π ∫ x(t) ∙ y(t) dt, where the limits of integration are from 0 to 10 (the bounds for t).

Substitute x(t) and y(t) into this formula and evaluate the integral. This requires advanced knowledge of calculus and can be quite complex, as it may involve integration techniques for exponential functions and polynomials, thus it is usually addressed at the college level.

To find the surface area, we need to set up the integral based on the equations given for x and y in terms of t. After the integral is set up, we evaluate it using the appropriate integration methods for the functions involved. This will give us the total surface area of the shape generated by the rotation.

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