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MA 2733 Midterm: MATH 2733 MSState Exam 1Solutions

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turquoisegnu128

15 Feb 2019

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Math 214-2 - fall 2000 - final exam (solutions) 1 = 2 {(81 27) (1 1)} = 108 . Compute the volume of revolution enclosed by gabriel"s horn. Gabriel"s horn: (arc length) find the length of the curve y = log (cos x) between x = 0 and x = /4. Y2 dx = z t t z t. 1: since log may be interpreted as either decimal logarithm or natural logarithm and the wording of the problem does not make clear how it must be interpreted here, both kinds of answers will be considered correct. 0 p1 + (y )2 dx = z /4. 0 s1 + tan2 x (ln 10)2 dx. 0 p1 + (y )2 dx =z /4 ds =z /4 sec x dx = log (sec x + tan x) /4. 4 ) log (sec 0 + tan 0) = log ( 2 + 1) log 1 = log ( 2 + 1)

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copperprairie-dog726

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