Mathematics: Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y=x^2, x=y^2 about y = - 6.

Use the method of cylindrical shells to find the volume

Ответ:

am2garcia5

30.01.2022

Mathematics

13.5

Step-by-step explanation:

We are given that two curves and a line

y=x^2,x=y^2 about y=-6

$x=\sqrt{y}$

Height= $\sqrt{y}-y^2$

Radius=y-(-6)=y+6

Now, we find intersecting points of two curves

y=(y^2)^2

y=y^4

y^4-y=0

y(y^3-1)=0

y=0

y^3-1=0

$y^3=1\implies y=1$

Now, volume generated by rotating the region by using cylindrical shell method is given by

$V=\int_{a}^{b}2\pi(height)(radius)dy$

Using the formula

$V=\int_{0}^{1}2\pi(\sqrt{y}-y^2)(y+6)dy$

$V=\int_{0}^{1}2\pi(y^{\frac{3}{2}}+6\sqrt{y}-y^3-6y^2)dy$

$V=2\pi[\frac{2}{5}y^{\frac{5}{2}}+4y^{\frac{3}{2}}-\frac{1}{4}y^4-2y^3]^{1}_{0}$

$V=2\pi(\frac{2}{5}+4-\frac{1}{4}-2)$

V=13.5

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Ответ:

kaylynn77

23.02.2020

Mathematics

The correct answer is:

576 sq. cm.

Explanation:

The surface area of a figure is the sum of the areas of the faces of the figure.

The slant height goes from the apex of the pyramid along a face and is perpendicular to the base at the bottom of that face. This gives us 2 measurements of each triangular face of the pyramid: the height (the slant height of the pyramid is the height of the triangular face), 18 cm, and the base of the triangular face, 12 cm.

The formula for the area of a triangle is A = 1/2bh; using our measurements we have:
A = 1/2(12)(18) = 108 sq. cm.

There are 4 triangular faces on a square pyramid, and each will have an area of 108 sq. cm.; this gives us:
4(108) = 432 sq. cm.

The area of the base is given by length times width, or 12*12 = 144 sq. cm.

This gives us a total surface area of 432+144=576 sq. cm.

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Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y=x^2, x=y^2 about y = - 6.

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