Answer to Question #148521 in Calculus for qwerty

Question #148521
Identify the curves (lines, parabola, etc.) generated by illustrating the graphs.

5. Consider the region enclosed by the curves y=√x, y=6-x, and the x-axis. Rotate this region about the x-axis and find the resulting volume.
6. Consider the region enclosed by the curves y=√x, y=6-x, and the x-axis. Rotate this region about the y-axis and find the resulting volume.
7. Consider the region enclosed by the curves y = x^2 − 2x and y = 3. Rotate this region about the line y = 3 and find the resulting volume.
8. Consider the region enclosed by the curves x = y^2 and x = 2 − y^2. Rotate this region about the line x = 3 and find the resulting volume.
1
Expert's answer
2020-12-04T12:50:15-0500
"Solution"

5. ) Is this the sum of two integrals or is it difference of two integrals?

First determine where the curves intersect: Obviously "y=\\sqrt{x}" meets the x-axis at "x=0" and "y=x-2" meets the x-axis at "x=2". Now "y=x-2" and "y=\\sqrt{x}" intersect when



"x-2=\\sqrt{x} \\implies x^2-4x+4=x \\implies x^2-3x+4=(x-4)(x+1)=0\\\\\n\\implies x=4, (not\\ x=-1)"



From the sketch above

Left: The region enclosed by the curves "y=\\sqrt{x},y=2-x," and the x-axis. When rotated about the x-axis, one region must be subtracted from the other. Instead of using representative rectangles, we simply indicate the appropriate radii with arrows.


From the sketch above sketch, if revolved about the x-axis will result in a solid that has been partially hollowed out (a cone has been removed). This requires a difference of integrals.


"V=Outer\\ volume-Inner \\ volume\\\\\n \\ = \\pi \\intop_0^4[\\sqrt x]^2dx-\\pi \\intop_2^4[x-2]^2dx\\\\\n \\ = \\pi \\intop_0^4 xdx-\\pi \\intop_2^4[x-2]^2dx\\\\\n\\ =\\frac{\\pi x^2}{2}|^4_0-\\frac{\\pi(x-2)^3}{3}|^4_2\\\\\n\\ = (8\\pi - 0)+(\\frac{8 \\pi}{3}-0)\\\\\n\\ = \\frac{16 \\pi}{3}"


6 ) Rotate the region about they-axis and find the resulting volume.


Is this the sum of two integrals or is it difference of two integrals?

Since the rotation is about the y-axis, the radii of the respective regions are horizontal, see the sketch below.



This is again a difference of two integrals.

Translating the curves into functions of "y" we have "x=y^2", "x=y+2", and "y=0" (the x-axis). The curves intersect the x-axis at "y=0". We’ve seen that the line and square root function meet when "x=4" since there, then they-coordinate of the intersection is "y=2".


"V=Outer\\ volume-Inner\\ volume=\\pi \\intop_0^2[y+2]^2dy-\\pi \\intop_0^2[y^2]^2dy\\\\\n\\ = \\frac{\\pi(y+2)^3}{3}|_0^2-\\frac{\\pi y^5}{5}|_0^2\\\\\n\\ = (\\frac{64 \\pi}{3}-\\frac{8 \\pi}{3})-(\\frac{32 \\pi}{3}-0)\\\\\n\\ = \\frac{184 \\pi}{15}"


7 ) The two curves meet when


"x^2-2x=3 \\implies x^2-2x-3=(x-3)(x+1)=0\\implies x=3,-1"

The parabola and the line are easy to sketch; see the below sketch on the left


Left: The region enclosed by the curves "y=x^2-2x" and "y=3". Since the axis of revolution is "y=3", a representative radius extends from the line "y=3" to the curve "y=x^2-2x". Right: The resulting solid of revolution about the line "y=3."


A representative radius extends from the line "y=3" to the curve "y=x^2-2x." The length of a radius is the difference between these two values, that is, the radius of a circular cross-section perpendicular to the line "y=3" is


"r=3-(x^2-2x)=3+2x-x^2"

Since a cross-section is a circle, its area is


"A=A(x)=\\pi r^2=\\pi(3+2x-x^2)^2"

Since we know the cross-sectional area, we find the volume

"V=\\intop_a^b A(x)dx=\\pi \\intop_{-1}^3(3+2x-x^2)dx\\\\\n\\ = \\pi \\intop_{-1}^3(9+12x-2x^2-4x^3+x^4)dx\\\\\n\\ = \\pi (9x+6x^2-\\frac{2x^3}{3}-x^4+\\frac{x^5}{5})|_{-1}^3\\\\\\\\\n\\ = \\pi[(27+54-18-81+\\frac{243}{5})-(-9+6+\\frac23-1-\\frac15)]\\\\\n\\ = \\frac{512 \\pi}{15}"


8. Consider the region enclosed by the curves "x = y^2" and "x = 2 \u2212 y^2."

The two curves meet when


"y^2=2-y^2 \\implies 2y^2=2 \\implies y=\\pm 1"

An ‘outer’ representative radius extends from the line "x=3" to the curve "x=y^2" and an ‘inner’ representative radius extends from the line "x=3" to "x=2-y^2". The length of a representative radius is the difference between the corresponding pairs of values


Left: The region enclosed by the curves "x=y^2" and "x=2-y^2" and two representative radii emanating from the axis of rotation "x=3." Right:The resulting ‘hollowed out’ solid of revolution about the line "x=3" looks like a ‘doughnut.’


The outer radius is "R=3-y^2" and the inner radius is "3-(2-y^2)=1+y^2." The integration will take place along the y-axis on the interval "[-1, 1]" because the disks are horizontal when points are rotated about the line "x=3." Since we know the cross-sections are circles, we can find the volume


"V=Outer\\ volume-Inner\\ volume=\\pi \\intop_{-1}^1[3-y^2]^2dy-\\pi \\intop_{-1}^1[1+y^2]^2dy\\\\\n\\ =\\pi \\intop_{-1}^19-6y^2+y^4\\ dy-\\pi \\intop_{-1}^11+2y^2+y^4\\ dy\\\\\n\\ =\\pi \\intop_{-1}^1 8-8y^2 dy\\\\\n\\ =\\pi(8y-\\frac{8y^3}{3})|_{-1}^1\\\\\n\\ = [(8- \\frac83)-(-8+\\frac83)]\\\\\n\\ = \\frac{32 \\pi}{3}"



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS