Explanation:Since the limits are constants and the integrand is 1: .
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2What does the double integral represent geometrically if ?
A.The area of the region
B.The volume of the solid below the surface and above the region
C.The mass of a lamina with density
D.The surface area of
Correct Answer: The volume of the solid below the surface and above the region
Explanation:A double integral of a positive function over a region calculates the volume under that surface bounded by .
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3Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Inner integral: . Outer integral: .
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4If limits of integration are constant, the order of integration:
A.Cannot be changed
B.Can be changed only if the function is symmetric
C.Can be changed arbitrarily without changing the limits
D.Can be changed, but limits must be inverted
Correct Answer: Can be changed arbitrarily without changing the limits
Explanation:For a rectangular region (constant limits) , provided is continuous.
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5Calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:Separable: .
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6Which of the following represents the area of a region using double integrals?
A.
B.
C.
D.
Correct Answer:
Explanation:The area of a region is given by the double integral of the function over that region.
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7Change the order of integration for .
A.
B.
C.
D.
Correct Answer:
Explanation:The region is bounded by . This is a triangle defined by . Changing order gives and .
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8In polar coordinates, the elementary area is replaced by:
A.
B.
C.
D.
Correct Answer:
Explanation:The Jacobian of the transformation from Cartesian to Polar coordinates is , so .
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9Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:. Then .
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10The value of the Jacobian for the transformation , is:
A.
B.
C.
D.
Correct Answer:
Explanation:.
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11Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:This represents the volume of a unit cube. , , . Product is 1.
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12The region of integration for is:
A.A square of side
B.A circle of radius
C.The upper semi-circle of radius
D.The first quadrant of a circle of radius
Correct Answer: The first quadrant of a circle of radius
Explanation:Limits are to and to . describes the upper circle, but and are both positive, so it is the first quadrant.
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13To change the order of integration in , the new limits for will be:
A. to
B. to
C. to
D. to
Correct Answer: to
Explanation:The region is and . This implies for going from to . So the inner integral becomes .
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14Evaluate by changing to polar coordinates.
A.
B.
C.
D.
Correct Answer:
Explanation:In polar: . Let , . Angle integral gives . Total .
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15For a triple integral , if is the sphere , which coordinate system is most convenient?
A.Cartesian coordinates
B.Cylindrical coordinates
C.Spherical coordinates
D.Curvilinear coordinates
Correct Answer: Spherical coordinates
Explanation:Spherical coordinates exploit the symmetry of a sphere, making limits constant ( to for ).
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16The Jacobian for the transformation from Cartesian to Cylindrical coordinates () is:
A.
B.
C.
D.
Correct Answer:
Explanation:The transformation is essentially polar in plane plus . The Jacobian is the same as polar coordinates: .
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17The Jacobian for the transformation from Cartesian to Spherical coordinates () is:
A.
B.
C.
D.
Correct Answer:
Explanation:The volume element in spherical coordinates is , so the Jacobian is .
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18Which integral gives the area of the region bounded by and ?
A.
B.
C.
D.
Correct Answer:
Explanation:Intersections are at . For in , . So limits for are to .
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19The volume of a solid region is given by:
A.
B.
C.
D.
Correct Answer:
Explanation:The volume of a 3D region is calculated by the triple integral of the function over that region.
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20When changing the order of integration for , the result is:
A.
B.
C.
D.
Correct Answer:
Explanation:The region is a square (). Limits remain constant and identical upon changing order.
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21Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Inner: . Outer: .
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22Calculate the area of the circle using double integrals.
A.
B.
C.
D.
Correct Answer:
Explanation:Area = . Here , so Area = . Using integral: .
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23What are the limits of for the region inside the circle ?
A. to
B. to
C. to
D. to
Correct Answer: to
Explanation:, so . The radius goes from origin () to the boundary ().
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24Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Volume of a box with sides 1, 2, 3. Volume = .
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25Dirichlet's integral is an extension of which function?
A.Sine function
B.Exponential function
C.Beta and Gamma functions
D.Logarithmic function
Correct Answer: Beta and Gamma functions
Explanation:Dirichlet's integrals for volumes involving over a tetrahedron are evaluated using Gamma functions.
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26Which of the following describes the volume of a cylinder of radius and height using cylindrical coordinates?
A.
B.
C.
D.
Correct Answer:
Explanation:Limits: , , . Integrand is Jacobian .
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27Calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:.
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28The region bounded by , , and is a:
A.Sphere
B.Cube
C.Tetrahedron
D.Cylinder
Correct Answer: Tetrahedron
Explanation:These four planes form a standard tetrahedron in the first octant.
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29What is the order of integration for ?
A. then then
B. then then
C. then then
D. then then
Correct Answer: then then
Explanation:The differentials appear as (or are implied by the limits: depends on ; depends on ; is constant).
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30In the double integral , if we change variables to and , the term is replaced by:
A.
B.
C.
D.
Correct Answer:
Explanation:The differential area element transforms using the absolute value of the Jacobian determinant: .
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31Identify the region of integration for .
A.Bounded by and
B.Bounded by and
C.Bounded by and
D.Bounded by and
Correct Answer: Bounded by and
Explanation:The limits explicitly state goes from the curve to for between 0 and 1.
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32Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:. Limits: . Then . Using reduction formula or identity: .
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33The volume of the sphere is calculated by . The result is:
A.
B.
C.
D.
Correct Answer:
Explanation:Standard formula for the volume of a sphere of radius .
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34The region for is a triangle with vertices:
A.
B.
C.
D.
Correct Answer:
Explanation:Lines are , (from lower limits) and (upper limit). Vertices are origin, x-intercept (1,0), and y-intercept (0,1).
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35To find the mass of a solid with density , one should evaluate:
A.
B.
C.
D.
Correct Answer:
Explanation:Mass is the integral of density over the volume.
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36Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Inner: . Outer: .
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37If and , then becomes:
A.
B.
C.
D.
Correct Answer:
Explanation:Integrand . Differential . Total integrand .
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38What is the Jacobian of the transformation ?
A.
B.
C.
D.
Correct Answer:
Explanation:.
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39In the change of order of integration for , the region is split into how many parts?
A.1 part
B.2 parts
C.3 parts
D.Does not need splitting
Correct Answer: 2 parts
Explanation:The upper boundary changes from a parabola to a line . When integrating first (horizontal strip), the right boundary definition changes depending on , requiring a split.
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40For a region defined by , the volume is calculated by:
A. where is
B.
C.
D.
Correct Answer: where is
Explanation:The volume under the surface is . The surface intersects at .
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41Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Area of rectangle width 1, height 2. .
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42Which integral calculates the volume of the solid bounded by and plane ?
A.
B.
C.
D.
Correct Answer:
Explanation:Volume under surface over the square .
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43In spherical coordinates, is equal to:
A.
B.
C.
D.
Correct Answer:
Explanation:By definition of spherical coordinates, is the distance from the origin.
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44If is the region bounded by , calculate .
A.
B.
C.
D.
Correct Answer:
Explanation:Current order : . Let . Result .
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45The limits for in cylindrical coordinates for a solid bounded by and paraboloid are:
A. to
B. to
C. to
D. to
Correct Answer: to
Explanation:In cylindrical coordinates, , so the upper surface is .
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46Evaluate .
A.Does not exist
B.Is a valid integral
C.Is invalid because limits depend on outer variables
D.Is invalid because is in the limit of but integrated after
Correct Answer: Is a valid integral
Explanation:Limits: depends on ; depends on ; is constant. Order matches dependencies. It is valid.
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47When calculating volume using triple integrals, if the density is constant , the mass is numerically equal to:
A.Surface Area
B.Volume
C.Density
D.Weight
Correct Answer: Volume
Explanation:. If , , which is Volume.
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48Double integrals can be used to calculate:
A.Area only
B.Volume only
C.Both Area and Volume
D.Neither Area nor Volume
Correct Answer: Both Area and Volume
Explanation: gives Area. gives Volume.
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49What is the relation between Beta () and Gamma () functions often used in evaluating multiple integrals?
A.
B.
C.
D.
Correct Answer:
Explanation:This is the fundamental relationship between Beta and Gamma functions.
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50Evaluate .
A.
B.
C.
D.
Correct Answer:
Explanation:Inner: . Outer: .
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