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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Evaluate the following integrals. 12-y p2-x-y TIT** y 0 0 xy dz dx dy
Compute the average value of the following functions over the region R. f(x, y) = 4 x - y; R = {(x, y): 0 ≤ x ≤ 2,0 ≤ y ≤ 2}
Evaluate the following integrals. o Jo V1-x² 2-x T 0 4yz dz dy dx
Evaluate the following integrals. 4 IST Jo Jo √x dz dx dy
Write an iterated integral of a continuous function f over the region R shown in the figure. YA 20 10 y = 2x R +4 4 (9, 18) y = 3x - 9 10 X
The surface of an island is defined by the following functions over the region on which the function is nonnegative. Find the volume of the island. N || 20 1 + x² + y² T x 2 M 11 20 1 + x² + y²
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. R X (1 + xy)² dA; R = {(x,
Evaluate the following integrals. In 8 Vz In 2y TY²³ In y ex+y²- dx dy dz
The surface of an island is defined by the following functions over the region on which the function is nonnegative. Find the volume of the island. z = 25 √x² + y² X z = 25-√x² + y²
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. R 6x³ex³ydA; R = {(x, y): 0
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. J y³ sin xy² dA; R = {(x,
Evaluate the following integrals. S.S. •√9-2² V₁+x²+z² 0 0 0 dy dx dz
Evaluate the following integrals. 77 0 TT 0 sin x sin y dz dx dy
The surface of an island is defined by the following functions over the region on which the function is nonnegative. Find the volume of the island. z = e(x² + y²)/8e-² x N z = e(x² + y²)/8e-2
The surface of an island is defined by the following functions over the region on which the function is nonnegative. Find the volume of the island. z = 100 - 4(x² + y²) X N z = 100 - 4(x² + y²)
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. JS x sec² xy dA; R = {(x,
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. [₁30 R 。-²-y²dA; R = {(x, y): x² + y² ≤ 9}
Evaluate the following integrals. 1.1.³ 1.² 1J0 6 4-2y/3 12-2y-3z 1 y dx dz dy
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. JI R R = {(x, y): x² + y² ≤ 4, x ≥ 0, y ≥ 0} 2 16 - x² - y² dA;
Evaluate the following integrals as they are written. TT/2 LYTTE [*'y cos. 0 0 y cos x³ dy dx
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. [(y + 1)ex(+¹) dA; R = {(x,
When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. y cos xy dA; R = {(x, y): 0
Evaluate the following integrals. 2V/16-y² 16-(x²/4)-y² -2V/16-y². 0 dz dx dy
Evaluate the following integrals. •√₁-2² VI-2 - y² [... 0 0 2xz dz dy dx
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. R 1 1 + x² +: dA; R = {(r, 0): 1 ≤ r ≤ 2,0 ≤ 0 ≤ πT} R
Evaluate the following double integrals over the region R. If (x³. — y³)² dA; R = {(x, y): 0 ≤ x ≤ 1, -1 ≤ y ≤ 1} R
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. JS 2xy dA; R = {(x, y): x² + y² ≤ 9, y ≥ 0} R
Evaluate the following integrals as they are written. px 1.5.² 0 0 2e-** dy dx
Evaluate the following integrals as they are written. In 2 2 [th et dy dx
Evaluate the following double integrals over the region R. JJ (x² (x² - y²)² dA; R = {(x, y): -1 ≤ x ≤ 2,0 ≤ y ≤ 1} R
Find the volume of the following solids using triple integrals.The region in the first octant bounded by the cone z = 1 - √x2 + y2 and the plane x + y + z = 1 N 1 y
Evaluate the following integrals. CVI-X Vi- S.S 0 0 0 VI-2 dz dy dx
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. [[ ² 2xy dA; R = {(r, 0): 1 ≤ r ≤ 3,0 ≤ 0 ≤ π/2} R
Evaluate the following double integrals over the region R. R ex+2y dA; R = {(x, y): 0 ≤ x ≤ ln 2, 1 ≤ y ≤ In 3}
Find the volume of the following solids using triple integrals.The wedge of the cylinder x2 + 4y2 = 4 created by the planes z = 3 - x and z = x - 3 ★ y
Sketch the given region of integration R and evaluate the integral over R using polar coordinates. J (x² + y²) dA; R = {(r, 0): 0 ≤ r ≤ 4,0 ≤ 0 ≤ 2} R
Evaluate the following integrals as they are written. V1-x² 0J-VI- 1-2²2 2x²y dy dx
Evaluate the following integrals as they are written. 8-1² -2x² x dy dx
Evaluate the following double integrals over the region R. J[. xy sin x² dA; R = {(x, y): 0 ≤ x ≤ √π/2,0 ≤ y ≤ 1} R
Find the volume of the following solids using triple integrals.The region between the sphere x2 + y2 + z2 = 19 and the hyperboloid z2 - x2 - y2 = 1, for z > 0 X y
Find the volume of the following solids using triple integrals.The region bounded by the surfaces z = ey and z = 1 over the rectangle {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ ln 2} In 21 X
Evaluate the following integrals as they are written. π/4 COS X LI -π/4J sin x dy dx
Evaluate the following double integrals over the region R. R y - X =dA; R = {(x, y); } ≤ x ≤ ³₁1 ≤ y ≤ 2}
Evaluate the following double integrals over the region R. X JS dA; R = {(x, y): 0 ≤ x ≤ 1,1 ≤ y ≤ 4} 20₂ R
Find the volume of the following solids using triple integrals.The region bounded by the parabolic cylinder y = x2 and the planes z = 3 - y and z = 0
Find the volume of the following solids. The solid bounded between the paraboloids z = 2x² + y² and z = 27x²2y² z = 27-x²-2y² z = 2x² + y² X Z R y
Evaluate the following integrals as they are written. 3 px+6 SS.** 0 J (x - 1) dy dx
Evaluate the following double integrals over the region R. D 4x³ cos y dA; R = {(x, y): 1 ≤ x ≤ 2,0 ≤ y ≤ π/2} R
Find the volume of the following solids. = The solid bounded between the paraboloids z z = 2x² - y² X N R x² + y² and z=2-x² - y² z = x² + y²
Find the volume of the following solids using triple integrals.The region bounded below by the coneand bounded above by the sphere 2 2 z = √x² + y²
Evaluate the following integrals as they are written. Sose 2x xy dy dx
Evaluate the following double integrals over the region R. JI (x² + xy) dA; R = {(x, y): 1 ≤ x ≤ 2, -1 ≤ y ≤ 1} R
Evaluate the following integrals as they are written. [56 0 X 6y dy dx
Find the volume of the following solids using triple integrals.The wedge above the xy-plane formed when the cylinder x2 + y2 = 4 is cut by the planes z = 0 and y = -z
Evaluate the following integrals as they are written. •2x 0/0, 15 xy² dy dx
Find the volume of the following solids using triple integrals.The prism in the first octant bounded by z = 2 - 4x and y = 8 N y
Find the volume of the following solids using triple integrals. The region in the first octant formed when the cylinder z = sin y, for 0 ≤ y ≤ π, is sliced by the planes y = x and x = 0 X z =
Evaluate the following double integrals over the region R. JS (x + 2y) dA; R = {(x, y): 0 ≤ x ≤ 3,1 ≤ y ≤ 4} R
Evaluate the following iterated integrals. 1 In 5 In 3 0 exty dx dy
Find the volume of the following solids using triple integrals.The region in the first octant bounded by the plane 2x + 3y + 6z = 12 and the coordinate planes (0, 0, 2) (6,0,0) (0, 4, 0) У 2x + 3y +
Evaluate the following iterated integrals. π/4 3 [³² 0 r sec 0 dr de
Evaluate the following iterated integrals. SS= y o 1 + x² 0 dx dy
Evaluate the following integrals. A sketch of the region of integration may be useful. [[[ D 0 ≤ y ≤ Vln 4,0 ≤z≤ 1} xyze -*-y²dV; D = {(x, y, z): 0 ≤ x ≤ Vln 2,
Evaluate the following integrals. A sketch of the region of integration may be useful. fff (xy + x2 + y2) dV; D = {(x, y, z): −1 ≤ x ≤ 1, xz D -2 ≤ y ≤ 2,-3≤z≤3}
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = {(x, y): 0 ≤ x ≤ 4, x² ≤ y ≤ 8√x}
Evaluate the following iterated integrals. pπ/2 1.².² 0 0 x cos xy dy dx
Evaluate the following iterated integrals. In 2 1h²2 1.6 Jo Jo 6xe³y dx dy
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx.R is the region in the first quadrant bounded by the y-axis and the
Evaluate the following integrals. A sketch of the region of integration may be useful. •π/2 [TT- 0 π/2 sin 7x cos y sin 2z dy dx dz
Evaluate the following integrals. A sketch of the region of integration may be useful. ISS 0 yze* dx dz dy
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx.R is the region in the first quadrant bounded by a circle of radius 1
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = {(x, y): 1 ≤ x ≤ 2₂ x + 1 ≤ y ≤ 2x + 4}
Evaluate the following iterated integrals. 2 LS (₂² (y² + y) dx dy
Evaluate the following integrals. A sketch of the region of integration may be useful. 0 In 4 In 3 0 0 In 2 e-x+y+z dx dy dz
Evaluate the following iterated integrals. 4 4 1 JO Vuv du dv
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = {(x, y): 0 ≤ x ≤ π/4, sin x ≤ y ≤ cos x }
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx.R is the triangular region with vertices (0, 0), (0, 2), and (1, 1).
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx. R = {(x, y): 0 ≤ x ≤ 2, 3x² ≤ y ≤ -6x + 24}
Evaluate the following iterated integrals. p3 pπ/2 0 x sin y dy dx
Evaluate the following integrals. A sketch of the region of integration may be useful. LITE -2√1 J1 xy² Z -dz dx dy
Sketch the following regions and write an iterated integral of a continuous function f over the region. Use the order dy dx.R is the triangular region with vertices (0, 0), (0, 2), and (1, 0).
Evaluate the following integrals. A sketch of the region of integration may be useful. -1-10 6xyz dy dx dz
Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R. -6 YA y = 2x + 24 УА 40 10 -2 R (4, 32) /y = 2x² + + 2 6 X
Evaluate the following iterated integrals. 0 -2 (2x + 3y) dx dy
Evaluate the following integrals. A sketch of the region of integration may be useful. -2√3J0 2 dx dy dz
Write two iterated integrals that equal R f(x, y) dA, where
Consider the regions R shown in the figures and write an iterated integral of a continuous function f over R. УА 10 2 y = 4x R 1 (2,8) y = x³ X
Evaluate the following iterated integrals. 2 (3x² + 4y3) dy dx
Evaluate the following iterated integrals. 2 S.S. 0 0 4xy dx dy
Evaluate the following iterated integrals. 2 [S²³² 0 x²y dx dy
How do you find the average value of a function over a region that is expressed in polar coordinates?
Describe and a sketch a region that is bounded on the left and on the right by two curves.
Write an iterated integral that gives the volume of a box with height 10 and base {(x, y): 0 ≤ x ≤ 5,-2 ≤ y ≤ 4}.
Consider the surfaces defined by the following equations.a. Identify and briefly describe the surface.b. Find the xy-, xz-, and yz-traces, if they exist.c. Find the intercepts with the three
Use Lagrange multipliers to find the maximum and minimum values of f (when they exist) subject to the given constraint. f(x, y, z) = x² + y² + z² subject to x² + y² + z² - 4xy = 1
Consider the surfaces defined by the following equations.a. Identify and briefly describe the surface.b. Find the xy-, xz-, and yz-traces, if they exist.c. Find the intercepts with the three
Consider the surfaces defined by the following equations.a. Identify and briefly describe the surface.b. Find the xy-, xz-, and yz-traces, if they exist.c. Find the intercepts with the three
Draw the regionWhy is it called a polar rectangle? {(r, 0): 1 ≤ r ≤ 2,0 ≤ 0 ≤ π/2}.
Write an iterated integral that gives the volume of the solid bounded by the surface f(x, y) = xy over the square R = {(x, y): 0 ≤ x ≤ 2,1 ≤ y ≤ 3}.
Find the values of ℓ and g with ℓ ≥ 0 and g ≥ 0 that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. U

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