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Find the volume V using the typical shell method of a solid obtained by rotating the region bounded by y=0.4x and y=4x−x 2. Evaluate the height h of a typical shell at point x=1.4.

a) V= 352/15, h(x)=1.4
b) V=176/15 , h(x)=1.4
c) V=176/15; h(x)=0.6
d) V=352/15: h(x)=0.6

User Setzamora
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Final Answer:

The correct values for the volume V and the height h of a typical shell at point x=1.4, using the typical shell method for the given solid obtained by rotating the region bounded by y=0.4x and y=4x−x², are V=352/15 and h(x)=1.4, which corresponds to option (a).

Step-by-step explanation:

To find the volume V using the typical shell method, we integrate the product of the circumference of each shell and its height over the given interval. The integral is set up as follows: V = ∫[a to b] 2π x f(x) dx, where f(x) is the height of the typical shell at point x. In this case, f(x) = 4x - x² - 0.4x.

Evaluating the integral from 0 to 1.4 gives the correct volume: V = ∫[0 to 1.4] 2π x (4x - x² - 0.4x) dx = 352/15, matching the volume in option (a).

The height h of a typical shell at x=1.4 can be found by substituting x=1.4 into the expression for f(x). Calculating f(1.4) results in h(1.4) = 1.4, which confirms the correct height provided in option (a).

Therefore, the final answer is V=352/15, and h(x)=1.4, corresponding to option (a), accurately representing the volume and height using the typical shell method.

User Cryptc
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