Let h(x,y)=2xy³i+4x²y³j. Calculate h(r) dr where C is the boundary of the triangula region in the first quadrant bounded by the x-axis, the line x-1 and the curve y = x².
Let h(x,y)=2xy³i+4x²y³j. Calculate h(r) dr where C is the boundary of the triangula region in the first quadrant bounded by the x-axis, the line x-1 and the curve y = x².
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![15.
16.
17.
a.
Let h(x,y)=2xy³i+4x³y²j. Calculate h(r) dr where C is the boundary of the triangular
region in the first quadrant bounded by the x-axis, the line x-1 and the curve y=x².
b.
Let g(x, y) = (2xy+e²-3)i + (x² - y² +sin y)j. Calculate g(r) dr where C is the ellipse
4x² +9y²=36
c. Use Green's Theorem to find the area enclosed by the asteroid
r(u) = cos' ui+sin'uj, 0≤u≤ 2
a. Find the area of the surface x+y+2=4 that lies within the cylinder x² + y² = 4
b. Calculate fxz do where S' is the portion of the plane z = 2x + 3y above the rectangle
1≤x≤2,1≤y≤3
a. Use the Divergence theorem to calculate (v-n)do for
v(x, y, z)=(x+z)i + (y+z)j + (x+2)k on the surface S: y² +22=1,0≤x≤4
b. Calculate the total flux of v(x, y, z)=2xi+xzj+z²k out of the solid bounded by the
paraboloid ==9-x² - y² and the xy-plane.
18. Use Stoke's theorem to find [[[(Vxv)•n] do given v(x, y, z) == i + xj + y k and S is the hemisphere
==√√4-x²-y². Take n as the upper unit normal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66f4da1e-847b-4b88-b6c9-6c1db3829f53%2F123889e1-c0d9-4330-a85f-0cb41ddb459f%2Fy1nqako_processed.png&w=3840&q=75)
Transcribed Image Text:15.
16.
17.
a.
Let h(x,y)=2xy³i+4x³y²j. Calculate h(r) dr where C is the boundary of the triangular
region in the first quadrant bounded by the x-axis, the line x-1 and the curve y=x².
b.
Let g(x, y) = (2xy+e²-3)i + (x² - y² +sin y)j. Calculate g(r) dr where C is the ellipse
4x² +9y²=36
c. Use Green's Theorem to find the area enclosed by the asteroid
r(u) = cos' ui+sin'uj, 0≤u≤ 2
a. Find the area of the surface x+y+2=4 that lies within the cylinder x² + y² = 4
b. Calculate fxz do where S' is the portion of the plane z = 2x + 3y above the rectangle
1≤x≤2,1≤y≤3
a. Use the Divergence theorem to calculate (v-n)do for
v(x, y, z)=(x+z)i + (y+z)j + (x+2)k on the surface S: y² +22=1,0≤x≤4
b. Calculate the total flux of v(x, y, z)=2xi+xzj+z²k out of the solid bounded by the
paraboloid ==9-x² - y² and the xy-plane.
18. Use Stoke's theorem to find [[[(Vxv)•n] do given v(x, y, z) == i + xj + y k and S is the hemisphere
==√√4-x²-y². Take n as the upper unit normal.
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