MATH 120 Midterm 2

Evaluate the double integral. ∫∫D xcos(y) dA where D is bounded by y = 0, y = x², and x = 1

(1 - cos1)/2

Evaluate the line integral ∫c y²z ds where C is the segment from (3, 1, 2) to (1, 2, 5).

(107√14)/12

Calculate the iterated integral. ∫₀¹∫₀¹ xy√(x² + y²) dy dx

(16√2 -2)/15

Calculate the iterated integral. ∫₁⁴∫₁² (x/y + y/x) dy dx

(21*ln2)/2

Evaluate the surface integral. ∫∫s y dS where S is the helicoid with vector equation r(u,v) = <ucosv, usinv, v>, 0≤u≤1, 0≤v≤π

(4 - 2√2)/3

Evaluate the line integral ∫c y ds where C is the curve given by x = t² and y = 2t and 0≤t≤3.

(40√10 - 4)/3

Evaluate the integral. ∫₀¹∫√x to 1 √(y³ + 1) dydx hint: the order of integration can be reversed.

(4√2 - 2)/9

Evaluate ∫c 2x ds where C is the piecewise curve including y = x² from (0, 0) to (1, 1) and the line segment from (1, 1) to (1, 2).

(5√5 - 1)/6 - 2

Evaluate the line integral ∫c xe^y ds where C is the line segment from (2, 0) to (5, 4).

(85e⁴ - 25)/16

Evaluate the integral. ∫₀¹∫ ³√(y) to 2 e^x⁴ dxdy hint: the order of integration can be reversed.

(e¹⁶ - 1)/4

Evaluate the integral. ∫₀¹∫3y to 3 e∧x² dxdy hint: the order of integration can be reversed.

(e⁹ - 1)/6

Evaluate the double integral. ∫∫D y/(x² + 1) dA where D = {(x, y) | 0≤x≤4, 0≤y≤√x}

(ln17)/4

Evaluate the line integral ∫c xe^yz ds where C is the segment from (0, 0, 0) to (1, 2, 3).

(√14 / 12)(e⁶ - 1)

Calculate the double integral. ∫∫R xsin(x + y) dA where R = [0, π/6] x [0, π/3]

(√3 -1)/2 - π/12

Show that the line integral is path independent and evaluate the integral. ∫c siny dx + (xcosy - siny) dy, C is any path from (2,0) to (1,π)

-2

Evaluate the line integral ∫c F dr where F(x, y, z) = xi + yj + xyk and C is given by the vector function r(t) = cost i + sint j + tk, 0 ≤ t ≤ π.

0

Evaluate the line integral ∫c F dr where F(x, y) = xy²i - x²j and C = r(t) = t³i + t²j (0 ≤ t ≤ 1).

1/20

Evaluate the line integral using Green's Thm. ∫c (y + e√x)dx + (2x + cosy²)dy C is the boundary of the region enclosed by the parabolas x=y² and y=x²

1/3

Evaluate the double integral. ∫∫D xy dA where D is enclosed by quarter circle y = √(1-x²) , x ≥ 0, and the axes.

1/8

Evaluate the surface integral. ∫∫s (x + y + z) dS where S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 - 2u + v, 0≤u≤2, 0≤v≤1

11√14

Evaluate the line integral. ∫c y²dx + x²ydy where C is the rectangle with vertices (0,0)(5,0)(5,4)(0,4)

120

Evaluate the given integral. ∫∫D x²y dA where D is the top half of the disk with center (0, 0) and radius 5.

1250/3

Find the area of the part of the surface z = 4 - 2x² + y that lies above the triangle with vertices (0,0),(1,0),(1,1)

13√2 / 12

Find the volume of the solid lying under the elliptic paraboloid x²/4 + y²/9 + z = 1 and above the rectangle R = [-1, 1] x [-2, 2]

166/27

Evaluate the surface integral. ∫∫s (xy²z) dS, where S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3]x[0,2]

171√14

Calculate the iterated integral. ∫-₃³∫₀->(π/2) (y + y²cosx) dx dy

18

Evaluate the line integral using Green's Thm. ∫c (1 - y³)dx + (x³ + e∧y²)dy C is the boundary of the region between the circles x² + y² = 4 and x² + y² = 9

195π/2

Calculate the iterated integral. ∫₁⁴∫₀² (6x²y - 2x) dy dx

222

Evaluate the double integral. ∫∫D (x² + 2y) dA where D is bounded by y = x, y = x³, and x ≥ 0.

23/84

Find the volume under the surface z = 1 + x²y² and above the region enclosed by x = y² and x = 4.

2336/27

Evaluate the line integral ∫c (2 + x²y) ds where C is the upper half of the unit circle.

2π + 2/3

Find the volume of the solid above the cone z = √(x² + y²) and below the sphere x² + y² + z² = 1.

2π/3 (1 - 1/√2)

Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,v) = sinu i + cosu sinv j + sinv k; u = π/6, v = π/6

2√3 x + 12 y - 6√3 z = √3

Find the volume of the solid inside the sphere x² + y² + z² = 16 and outside the cylinder x² + y² = 4.

32π/3

Evaluate the line integral ∫c z²dx + y²dz + x²dy where C is the segment from (1, 0, 0) to (4, 1, 2).

35/3

Find an equation of the tangent plane to the given parametric surface at the specified point. x = u + v, y = 3u², z = u - v (2, 3, 0)

3x - y + 3z = 3

Evaluate the given integral. ∫∫R arctan(y/x) dA where R = {(x, y) | 1≤ x² + y² ≤ 4, 0 ≤ y ≤ x}

3π²/64

Find the area of the part of the plane 3x + 2y + z = 6 that lies in the first octant.

3√14

Find the area of the surface with parametric equations x = u², y = uv, and z = 1/2 v², 0 ≤ u ≤ 1, 0 ≤ v ≤ 2

4

Evaluate the double integral. ∫∫D y√(x² + y²) dA where D = {(x, y) | 0≤x≤2, 0≤y≤x}

4/3

Show that the line integral is path independent and evaluate the integral. ∫c 2xe^-y dx + (2y - x²e^-y)dy, C is any path from (1,0) to (2,1).

4/e

Find the volume of the solid under the plane 2x + y + z = 4 and above the disk x² + y² = 1.

4π

Compute the surface integral ∫∫s x² dS, where S is the unit sphere x² + y² + z² = 1. Hint: parametrize a sphere using spherical coordinates: x = sinΩcosθ, y = sinΩsinθ, z = cosΩ

4π/3

Find the area of the part of the plane with vector equation r(u,v) = <u + v, 2 - 3u, 1 + u - v> that is given by 0 ≤ u ≤ 2, -1 ≤ v ≤ 1

4√22

Evaluate the line integral ∫c F dr where F(x, y, z) = sinx i + cosy j + xz k and C is given by the vector function r(t) = t³i - t²j + tk, 0 ≤ t ≤ 1.

6/5 - cos1 - sin1

Find the volume of the solid under the paraboloid z = x² + y² and above the disk x² + y² = 25.

625π/2

Find the volume of the solid enclosed by the paraboloid z = 2 + x² + (y - 2)² and the planes z = 1, x = ±1, y = 0, and y = 4.

64/3

Calculate the iterated integral. ∫₀¹∫₀¹ (x + y)² dx dy

7/6

Evaluate the line integral ∫c F dr where F(x, y, z) = (x + y²)i - xzj + (y + z)k and C = r(t) = t²i + t³j - 2tk (0 ≤ t ≤ 2).

8

Find the volume of the solid lying enclosed by the surface z = 1 + x²ye∧y and the planes z = 0, x = ±1, y = 0, and y = 1

8/3

Evaluate the given integral. ∫∫D x dA where D is the region in the first quadrant that lies between circles x² + y² = 4 and x² + y² = 2x.

8/3 - π/2

Evaluate the line integral ∫c x²dx + y²dy where C consists of the arc of the circle x² + y² = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3).

83/3

Calculate the double integral. ∫∫R xy²/(x² + 1) dA where R : {(x,y) | 0≤x≤1, -3≤y≤3}

9ln2

Calculate the double integral. ∫∫R 1/(1 + x + y) dA R = [1, 3], [1, 2] hint: ∫lnxdx = xlnx - x + C

9ln3 - 5ln5 -2ln2

Find the volume of the solid bounded by the paraboloid z = 1 + 2x² + 2y² and the plane z = 7 in the first octant.

9π/4

Determine whether or not the vector field is conservative. If conservative, find a function f s.t. F = ∨f F(x, y, z) = e^yz i + xze^yz j + xye^yz k

F is conservative. f(x, y, z) = xe^yz + K

Determine whether or not the vector field is conservative. If conservative, find a function f s.t. F = ∨f F(x, y, z) = y²z³i + 2xyz³j + 3xy²z²k

F is conservative. f(x, y, z) = xy²z³ + K

Find the gradient vector field of f. f(s, t) = √(2s + 3t)

F(s, t) = <1/√(2s + 3t), 3/2√(2s + 3t)>

Find the gradient vector field of f. f(x, y) = y sin xy

F(x, y) = <y² cos xy, sin xy + xy cos xy>

Find the gradient vector field of f. f(x, y, z) = x²e^(y/z)

F(x, y, z) = <2xye^(y/z), x²e^(y/z) + (x²ye^(y/z))/z, (-x²y²e^(y/z))/z²>

Find the flux of F across S. Use positive orientation. F(x, y, z) = ze^xy i - 3ze^xy j + xy k x = u + v, y = u - v, z = 1 + 2u + v 0 ≤ u ≤ 2, 0 ≤ v ≤ 1

Flux = 4

Find the flux of F across S. Use positive orientation. F(x, y, z) = xy i - yz j + zx k S is part of the paraboloid z = 4 - x² - y² that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 * integrate first with respect to y

Flux = 713/180

Use Lagrange Multipliers to find the extreme values of the function subject to the given constraint. f(x,y) = xy; 4x² + y² = 8

Maximum of 2 at (1, 2) and (-1, -2) Minimum of -2 at (-1, 2) and (1, -2)

Use Lagrange Multipliers to find the extreme values of the function subject to the given constraint. f(x,y,z) = 2x + 2y + z; x² + y² + z² = 9

Maximum of 9 at (2, 2, 1) Minimum of -9 at (-2, -2, -1)

Use Lagrange Multipliers to find the extreme values of the function subject to the given constraint. f(x,y,z) = e^(xyz); 2x² + y² + z² = 24

Maximum of e¹⁶ at (2, 2√2, 2√2), (-2, -2√2, 2√2), (2, -2√2, -2√2), (-2, 2√2, -2√2) Minimum of 1/e¹⁶ at (-2, 2√2, 2√2), (2, -2√2, 2√2), (2, 2√2, -2√2), (-2, -2√2, -2√2)

Use Lagrange Multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = x² + y² + z²; x⁴ + y⁴ + z⁴ = 1

Maximum of √3 at (±⁴√(1/3), ±⁴√(1/3), ±⁴√(1/3)) Minimum of 1 at (0, 0, ±1), (0, ±1, 0), (±1, 0, 0)

Determine whether or not F is a conservative vector field. If conservative, find a function f s.t. F = ∨f. F(x, y) = (xy + y²)i + (x² + 2xy)j

Not conservative

Line integrals of conservative vector fields are:

Path independent

a. Find a function f such that F = ∨f b. use part a to evaluate ∫c F dr along the curve C. F(x, y, z) = siny i + (xcosy + cosz)j - ysinz k C: r(t) = sint i + t j + 2t k, 0 ≤ t ≤ π/2.

a. f(x, y, z) = xsiny + ycosz b. 1 - π/2

a. Find a function f such that F = ∨f b. use part a to evaluate ∫c F dr along the curve C. F(x, y, z) = yzi + xzj + (xy + 2z)k C is the line segment from (1, 0, -2) to (4, 6, 3).

a. f(x, y, z) = xyz + z² b. 77

Identify the surface with the given vector equation. r(u, v) = <scost, ssint, s>

cone x² + y² = z²

Compute curl and divergence of the vector field. F(x, y, z) = xy²z²i + x²yz²j + x²y²zk

curl F = 0 div F = y²z² + x²z² + x²y²

Compute curl and divergence of the vector field. F(x, y, z) = xye^z i + yze^x k

curl F = ze^x i + (xye^z - yze^x)j - xe^z k div F = ye^z + ye^x

If F = Pi + Qj + Rk is a vector field on R³ and P, Q, and R have continuous 2nd order partial derivatives, then: div curl F = 0

don't just move on from this, it's important

If F is a conservative vector field, curl F = 0. Conversely, if f(x, y, z) is a function of three variables with continuous second order partial derivatives, then curl(gradient f) = 0.

don't just move on from this, it's important

Evaluate the line integral ∫c e^x dx where C is the curve y = x³ from (-1, -1) to (1, 1).

e - 1/e

Calculate the iterated integral. ∫₀¹∫₀² ye^(x-y) dx dy

e² - 2e -1 +2/e

Determine whether or not F is a conservative vector field. If conservative, find a function f s.t. F = ∨f. F(x, y) = (lny + y/x)i + (lnx + x/y)j

f(x, y) = xlny + ylnx + K

Determine whether or not F is a conservative vector field. If conservative, find a function f s.t. F = ∨f. F(x, y) = (y² - 2x)i + 2xyj

f(x, y) = xy² - x² + K

Determine whether or not F is a conservative vector field. If conservative, find a function f s.t. F = ∨f. F(x, y) = (2xy + y⁻²)i + (x² - 2xy⁻³)j

f(x, y) = x²y + xy⁻² + K

vector equation of a plane containing a point a and two vectors b₁ and b₂

r(u, v) = a + ub₁ + vb₂

Remember: parametrizing a surface (like a cone or cylinder) will involve an extra r in the integral (check #6 on pset 10).

read that one more time

Evaluate the integral. ∫₀¹∫x² to 1 √(y)*sin(y) dydx hint: the order of integration can be reversed.

sin1 - cos1

Identify the surface with the given vector equation. r(u, v) = (u + v)i + (3 - v)j + (1 + 4u + 5v)k

the plane 4x - y - z = -4

Find a parametric representation for the plane that passes through the point (0, -1, 5) and contains the vectors <2, 1, 4> and <-3, 2, 5>

x = 2u - 3v, y = -1 + u + 2v, z = 5 + 4u + 5v no restrictions on u, v.

Find a parametric representation for the part of the cylinder x² + z² = 9 that lies above the xy plane and between the planes y = -4 and y = 4.

x = 3cosθ, y = y, z = 3sinθ 0 ≤ θ ≤ π, -4 ≤ y ≤ 4

parametric equation of a sphere

x = a sinΩ cosθ, y = a sinΩ sinθ, z = a cosΩ

Find a parametric representation for the part of the plane z = x + 3 inside the cylinder x² + y² = 1.

x = rcosθ, y = rsinθ, z = rcosθ + 3 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1.

Use a double integral to find the area of the region. One loop of the rose r = cos3θ

π/12

Use Greens Thm to evaluate ∫c Fdr. Check the curve's orientation before computing. F(x, y) = <e^-x + y², e^-y + x² C consists of the arc of the curve y = cosx from (-π/2,0) to (π/2,0) and the line segment from (π/2,0) to (-π/2,0)

π/2

Use a double integral to find the area of the region. The region inside the cardioid r = 1 + cosθ and ourside the circle r = 3cosθ

π/4

Use Greens Thm to evaluate ∫c Fdr. Check the curve's orientation before computing. F(x, y) = <√(x² + 1), tan⁻¹x> C is the triangle from (0,0) to (1,1) to (0,1) to (0,0).

π/4 - ln2/2

Find the area of the part of the plane x + 2y + 3z = 1 that lies inside the cylinder x² + y² = 3

π√14

Evaluate the surface integral. ∫∫s (x² + y²) dS where S is the surface given by r(u,v) = <2uv, u² - v², u² + v²>, u² + v² ≤1

π√2

Find the area of the part of the cone z = √(x² + y²) between the plane y = x and the cylinder y = x²

√2 /6

Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,v) = ucosv i + usinv j + v k; u = 1, v = π/3

√3/2 x - 1/2 y + z = π/3

Find the flux of F across S. F(x, y, z) = -x i - y j + z³ k S: z = √(x² + y²) between z = 1 and z = 3

∫∫s FdS = 1712/15 π