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MATH 265 Midterm: MATH 265 Iowa State m3Practice

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turquoisegnu128

15 Feb 2019

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Iowa State University

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0 ez dz dy dx: set up an integral in polar coordinates that nds the volume above the xy-plane, below the plane z = 10 2x + 3y and between the cylinders x2 + y2 = 1 and x2 +y2 = 4. Evaluate the integral to nd the total volume: evaluate the integral. Change the order of integration to dx dy dz: consider the planar region bounded between the lines x = 1 and x = 4 and inside the parabola x = y2. Given that the density at a point (x, y) is (x, y) = xy2. 0 z 4 y/2 integration. ) sin(x2) dx dy. (hint: change order of: the archimedean spiral is described by the curve r = (in polar coordinates). For the homoge- nous volume (i. e. , density is constant) bounded be- low by the xy-plane, bounded above by the surface z = x2 + y2 and over the region from the origin to the.

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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

Solve all please.

INSTRUCTIONS: There are 15 problems on this exam, and each problem is worth 13 points, plus 5 points for writing down an approximate volume of the planet Earth in cubic miles at the bottomleft corner of page 3, for a total possible 200 points. For this exam, you are allowed to use one 8.5"X11" sheet of notes (both sides, in your own handwriting), pencil, eraser, ruler, scratch paper and calculator. Even though the questions are multiple-choice, you need to show enough work to justify your answer in order to receive full credit. After you have determined the correct solution to each problem, write the letter corresponding to your answer in the appropriate space on the right side of the page AND circle your answer. If your answer does not agree with any of the choices, after double-checking your work, choose option E) and prominently mark your answer (box, circle, underline, etc.). Time limit: Two hours. Attach (i.e. staple) any notecard and scratch paper to the rear of this exam paper. Good luck! MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. The northern third of Indiana is a rectangle measuring 96 miles by 132 miles. Thus, let D = [0, 96] times [0,132]. Assuming that the total annual snowfall (in inches), S(x,y), at (x,y) euro D is given by the function S(x,y) = 60e-0.001 (2x + y) with (x,y) euro D, find the average snowfall on D. 51.14 inches 52.06 inches 51.78 inches 52.44 inches Evaluate the work done between point 1 and point 2 for the conservative field F. Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. Evaluate the surface integral of G over the surface S. S is the portion of the cone Find the flux of the vector field F across the surface S in the indicated direction. Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. Integrate the function f over the given region. f(x,y) = 1/ln x over the region bounded by the x-axis, line x = 9, and curvey y = ln x Reverse the order of integration and then evaluate the integral Use a double integral in polar coordinates to find the area of the region specified in polar coordinates. the region inside both r = 7 sin Theta and r = 7 cos Theta Use spherical coordinates to find the volume of the indicated region. the region bounded above by the sphere x2 + y2 + z2 = 25 and below by the cone z = Find the center of mass of the rectangular solid of density (x,y,z) = xyz defined by 0 LE x LE 8, Use the given transformation to evaluate the integral. where R is the parallelogram bounded by the lines y = 8x + 8, y = 8x +9, y = -6x + 2, y = -6x +7 Evaluate the line integral along the curve C. Find the work done by F over the curve in the direction of increasing t.

please show all steps.

Convert the above Riemann Sum to an integral by letting delta x rightarrow 0. Since this integral must be zero, we get an equation. Solve the equation for x to show that Suppose that the length of the rod is 5m and that its density is delta (x ) = (0.lx3 + 2) kg/3. Find the center of mass of the rod, and indicate where it is on a diagram. Is your center of mass to the left or right of the geometric center of the rod? Explain why this is so. The same process can be used to compute the center of mass of a 2-D object. The coordinates of the center of mass of such an object will be and . The notation used to refer to the different quantities is (the moment about the y/-axis), (the moment about the x -axis), and . Of course, M is simply the total mass of the object, represents the "vertical" line upon which the solid would balance, and represents the "horizontal" line upon which the solid would balance Therefore, the point (x, y) represents the one point at which the solid would balance perfectly. An exactly analogous set of formulas holds for 3-D objects. Find the center of mass of a lamina defined by the parabola y = 2 - 3x2 and the x-axis if its density is given by delta(x, y ) = x4 y2. For the above region, find the geometric center, and if it differs from the center of mass, explain why this is so and why it makes sense. (Hint: making a judicious choice for delta will ensure that the geometric center and the center of mass are the same point - so finding the center of mass with the above formulas and this choice of delta will actually give you the location of the geometric center of the object.) When engineers design an appliance, one of the concerns is how hard the appliance will be to tip over, when tipped, it will right itself as long as its center of mass lies on the correct side of the fulcrum, the point on which the appliance is riding as it tips. See the figure below. Suppose the profile of an appliance of approximately constant density is parabolic (y = a ( l - x2)), like an old-fashioned radio. What values of a will guarantee that the appliance will have to be tipped more than 45 degree to fall over? Your answer should be exact.