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solong [7]
2 years ago
6

Region R is bounded by the curves y = 4x^2 and y = 4. A solid has base R, and cross sections perpendicular to the y-axis are sem

icircles with the diameter lying in R. The volume of this solid is
A. pi/4
B. pi/2
C. pi
D. 2pi

Mathematics
1 answer:
vlada-n [284]2 years ago
6 0

Each cross section has a diameter equal the horizontal distance between the two "halves" of the parabola. We have

y = 4x²   ⇒   x = ±√y/2

so the diameter for some given y would be √y/2 - (-√y/2) = √y.

The volume of a cross section with thickness ∆y is then

π/2 (√y/2)² ∆y = π/8 y ∆y

The total volume of the solid would then be

\displaystyle \int_0^4 \frac\pi8 y \, dy = \frac\pi8 \times \frac12 y^2 \bigg|_{y=0}^{y=4} = \boxed{\pi}

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We can substitute y for r(x), to write in the notation:

r(x)=\frac{x-1}{x+4}\\y=\frac{x-1}{x+4}

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We want to find x-intercepts here, so we substitute 0 into y and solve for x. Shown below:

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<u>The only x-intercept is x = 1</u>

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