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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

If possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume that f is continuous on the region. 2π
If possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume that f is continuous on the region.
Geographers measure the geographical center of a country (which is the centroid) and the population center of a country (which is the center of mass computed with the population density). A
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x= v₁y = u
Use spherical coordinates to find the volume of the following regions.A ball of radius a > 0
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T: x=-v, y = u
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Compute the Jacobian J(u, v) of the following transformations. T: x = u + v, y = u - v
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:x= (u + v)/2, y = (u - v)/2
Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming
Let S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. T:xu, y = 2v + 2
Compute the Jacobian J(u, v) of the following transformations. T:x=u²v², y = 2uv
Compute the Jacobian J(u, v) of the following transformations. T: x = 4uv, y = -2u + 3v
If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration.The region above the cone z = r and below the
Compute the Jacobian J(u, v) of the following transformations. T: x= 3u, y = 2v + 2
Choose the best coordinate system and find the volume of the following solid regions. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may
To evaluate the following integrals carry out the following steps.a. Sketch the original region of integration R and the new region S using the given change of variables.b. Find the limits of
Choose the best coordinate system and find the volume of the following solid regions. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may
To evaluate the following integrals carry out the following steps.a. Sketch the original region of integration R and the new region S using the given change of variables.b. Find the limits of
Evaluate the following integrals using a change of variables of your choice. Sketch the original and new regions of integration, R and S. JI y4 dA; R is the region bounded by the hyperbolas xy = 1
Before a gasoline-powered engine is started, water must be drained from the bottom of the fuel tank. Suppose the tank is a right circular cylinder on its side with a length of 2 ft and a radius of 1
Evaluate the following integrals using a change of variables of your choice. Sketch the original and new regions of integration, R and S. JS (y² + xy - 2x²) dA; R is the region bounded by the
Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the
Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the
Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the
Use a change of variables to evaluate the following integrals. III yz dV; D is bounded by the planes x + 2y = 1, x + 2y = 2, D x -z = 0, x-z = 2, 2yz = 0, and 2y -z = 3.
Find the average value of the following functions on the given curves. f(x, y) = xe on the unit circle centered at the origin
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Use a change of variables to evaluate the following integrals. x dV; D is bounded by the planes y - 2x = 0, y 2x = 1, D z - 3y = 0, z - 3y = 1,2 - 4x = 0, and z 4x = 3.
Consider the following regions R and vector fields F.a. Compute the two-dimensional divergence of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State
Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the
Find the average value of the following functions on the given curves. f(x, y) = √4 + 9y²/3 on the curve y = x³/2, for 0 ≤ x ≤ 5
Sketch a few representative vectors of the following vector fields. F = (x, y, z) √x² + y² + 2²
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Sketch a few representative vectors of the following vector fields. F = (y, -x, 0)
Find the average value of the following functions on the given curves. f(x, y) = x² + 4y² on the circle of radius 9 centered at the origin
Find the average value of the following functions on the given curves. f(x, y) = x + 2y on the line segment from (1, 1) to (2,5)
Use a line integral on the boundary to find the area of the following regions.The region bounded by the curve r(t) = (t(11²), 1-²), for -1 ≤ t ≤ 1
Explain with pictures what is meant by a simple curve and a closed curve.
Sketch a few representative vectors of the following vector fields. F = (x, y, z)
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Sketch a few representative vectors of the following vector fields. F = (1, 0, z)
Use a line integral on the boundary to find the area of the following regions.The region bounded by the parabolas r(t) = (t, 21²) and r(t) = (t, 12 - 1²), for-2≤ t ≤2
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
Use a line integral on the boundary to find the area of the following regions. {(x, y): x²/25 + y²/9 ≤ 1}
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
Use a line integral on the boundary to find the area of the following regions. {(x, y): x² + y² ≤ 16}
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Use a line integral on the boundary to find the area of the following regions.A region bounded by an ellipse with semimajor and semiminor axes of length 6 and 4, respectively.
Use a line integral on the boundary to find the area of the following regions.A disk of radius 5
Match vector fields a–d with graphs A–D. a. F = c. F = (0, x²) (2x, -y) b. F= (xy, x) d. F = (y,x)
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State whether
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Sketch the following vector fields. F = (e ³, 0)
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let R* and D* be open regions of R2 and R3, respectively, that do not
Sketch the following vector fields. F = X y 2 x +y Vx? +y? Vx
a. Find a parametric description for C in the form r(t) = (x(t), y(t)), if it is not given.b. Evaluate |r'(t)|.c. Convert the line integral to an ordinary integral with respect to the parameter and
Evaluate the following line integrals. [.x² x²y ds; C is the line r(s) = (s/V2, 1-s/√2), for 0 ≤S≤ 4.
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency. c. State
Determine whether the following vector fields are conservative on R2. F = (2x³ + xy², 2y³ + x²y)
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State whether
Sketch the following vector fields. F = (x, y - x)
Evaluate the following line integrals. √(x²- (x²2y2) ds; C is the line r(s) = (s/√2, s/√2), for 0 ≤S≤ 4.
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State whether
Determine whether the following vector fields are conservative on R2. F = (-y, x + y)
Determine whether the following vector fields are conservative on R2. F = (e cos y, e* sin y)
Sketch the following vector fields. F = (x + y, y)
Evaluate the following line integrals. xy ds; C is the unit circle r(s) = (cos s, sin s), for с 0 ≤S≤ 2TT.
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State whether
Consider the following regions R and vector fields F.a. Compute the two-dimensional curl of the vector field.b. Evaluate both integrals in Green’s Theorem and check for consistency.c. State whether
Determine whether the following vector fields are conservative on R2. F = (-y, -x)
Sketch the following vector fields. F = (y₂-x)
Sketch the following vector fields. F = (2x, 3y)
Consider the radial fielda. Explain why the conditions of Green’s Theorem do not apply to F on a region that includes the origin.b. Let R be the unit disk centered at the origin and computec.
Determine whether the following vector fields are conservative on R2. F = (x, y)
Sketch the following vector fields. F = (x, y)
Sketch the following vector fields. F = (-x, -y)
Determine whether the following vector fields are conservative on R2. F = (1, 1)
Consider the radial fielda. Verify that the divergence of F is zero, which suggests that the double integral in the flux form of Green’s Theorem is zero.b. Use a line integral to verify that the
Determine whether ∫C F • dr along the paths C1 and C2 shown in the following vector fields is positive or negative. Explain your reasoning. to // // YA 3 14 4-2 3+ -3 C₂ 3 X
Determine whether ∫C F • dr along the paths C1 and C2 shown in the following vector fields is positive or negative. Explain your reasoning. -4 YA 2 X
Sketch the oriented quarter circle from (1, 0) to (0, 1) and supply a parameterization for the curve. Draw the unit normal vector (as defined in the text) at several points on the curve.
Discuss one of the parallels between a conservative vector field and a source free vector field.
Consider the following vector fields F and closed oriented curves C in the plane (see figures).a. Based on the picture, make a conjecture about whether the circulation of F on C is positive,
Consider the following vector fields F and closed oriented curves C in the plane (see figures).a. Based on the picture, make a conjecture about whether the circulation of F on C is positive,
How do you calculate the flux of a two-dimensional vector field across a smooth oriented curve C?
R3 Given the force field F, find the work required to move an object on the given oriented curve. F = (x, y, z) (x² + y² + z²)3/2 (10, 10, 10) on the line segment from (1, 1, 1) to
Sketch a two-dimensional vector field that has zero divergence everywhere in the plane.
Given a two-dimensional vector field F and a smooth oriented curve C, what is the meaning of the flux of F across C?
Sketch a two-dimensional vector field that has zero curl everywhere in the plane.

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