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VikaD [51]
3 years ago
13

Let R be the region in the first quadrant bounded by the curves y = x^3 and y = 2x - x^2. Calculate the following quantities.

Mathematics
1 answer:
daser333 [38]3 years ago
4 0
(a)

2x - x^2 \ge x^3 \\ \\
A = \int_0^1(2x - x^2 - x^3) dx = \left[ x^2  - \frac{1}{3}x^3 - \frac{1}{4}x^4\right]_0^1  \\ \\
A = 1 - \frac{1}{3} - \frac{1}{4} = \frac{5}{12}

(b) A cross-section is a washer with inner radius x^3 and outer radius 2x - x^2 so its area is \pi(2x - x^2)^2 - \pi(x^3)^2.

V = \int_0^1 A(x) dx = \int_0^1 \pi \left[ (2x - x^2)^2 - (x^3)^2\right] dx \\ \\
= \int_0^1 \pi( 4x^2 - 4x^3 + x^4 - x^6)dx \\ \\
=\pi \left[\frac{4}{3}x^3 - x^4 + \frac{1}{5}x^5 - \frac{1}{7}x^7 \right]_0^1 = \pi \left(\frac{4}{3} - 1 + \frac{1}{5}- \frac{1}{7} \right) \\
\\
=\frac{41}{105}\pi

(c) Using the method of cylindrical shells,

V = \int_0^1 2\pi x(2x - x^2 - x^3)dx = \int_0^1\pi (2x^2 - x^3 - x^4)dx \\ \\
= 2\pi \left[\frac{2}{3}x^3 - \frac{1}{4}x^4 - \frac{1}{5}x^5 \right]_0^1 \\
\\
= 2\pi \left( \frac{2}{3} - \frac{1}{4} - \frac{1}{5}\right) = \frac{13}{30}\pi
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