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APPM 1360 Midterm: appm1360summer2014exam2_sol

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turquoisegnu128

31 Jan 2019

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Summer 2014: (a) (12 pts) let r be the region between the curve y = 1 x, the x-axis and the y-axis. Determine the volume of the solid whose cross sections in r, perpendicular to the y axis, are squares. Be sure to evaluate the integral(s) involved. (b) (12 pts) a cup is formed by rotating the region between y = 0, y = 9, x = 0 and the line y = 2x 1 about the y axis. Using cylindrical shells, set up but do not evaluate the integral(s) representing the volume of the cup. Solution: (a) since we are working with cross-sections perpendicular to the y-axis, we will use dy. The area for a square is a = l2. For this problem, l will be the distance from the y axis to the curve y = 1 x or x = 1 y2.

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Related Questions

I need help solving some problems including area bounded by aregion and volume by cylindrical shells.

1) y = 3/x

y=12x

y=1/12x

x>0

2)
Find thevolume V of the solid obtained by rotating the regionbounded by the given curves about the specifiedline.

y=4-(1/2x)

y=0

x=2

x=3

rotated about xaxis

3)Find thevolume V of the solid obtained by rotating the regionbounded by the given curves about the specified line.
x = 4sqrt(5y)

x=0

y=3

rotated on y axis

4)Find the volume V of the solid obtained by rotatingthe region bounded by the given curves about the specifiedline.
y = ln3x

y =4

y =5

x = 0

rotated on yaxis

5)
Find the volume V of the solid obtained by rotating theregion bounded by the given curves about the specified line.

y =3x⁵

y =3x

x Ã¢â°Â¥ 0

rotated on X axis

6)
Find the volume V of the solid obtained by rotating theregion bounded by the given curves about the specifiedline.

y² = 2x
x = 2y

about the y-axis

7)A CAT scan produces equally spaced cross-sectional views of ahuman organ that provide information about the organ otherwiseobtained only by surgery. Suppose that a CAT scan of a human livershows cross-sections spaced 1.5 cm apart. The liver is 15 cm longand the cross-sectional areas, in square centimeters, are 0,17, 58, 79,95, 107, 118, 129, 64, 39,and 0. Use the Midpoint Rule with n = 5 to estimate thevolume V of the liver.

8)
Let S be the solid obtained by rotating the region shown inthe figure about the y-axis. (Assume a = 7 and b = 2.)
y=ax(x-b)^2
What are its circumference c and height h?
Use shells to find the volume V of S.

9) Let S be the solid obtained by rotating the region shownin the figure about the y-axis. (Assume a = 4 andb = 2.)
y=asin(bx^2)

What are its circumference c andheight h?
Use shells to find the volume V of S.

10)Use the method of cylindrical shells to find the volumegenerated by rotating the region bounded by the given curves aboutthe y-axis.

y = 2x²
y = 12x Ã¢Ëâ 4x^2

11)Use the method of cylindrical shells to find thevolume of the solid obtained by rotating the region bounded by thegiven curves about the x-axis.

xy = 3
x =0
y =3
y =5

12)
Use the method of cylindrical shells to find the volume V ofthe solid obtained by rotating the region bounded by the givencurves about the x-axis.

x + y = 5
x = 9 Ã¢Ëâ (y Ã¢Ëâ 2)²

13)Use the method of cylindrical shells to find the volume Vgenerated by rotating the region bounded by the given curves aboutthe y-axis.

y = 5x²
y = 0
x = 1