In Exercises 9-22, change the Cartesian
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- Say that you need to compute a double integral of the function f(x,y)=xy over the region D bounded by the x-axis, y=x, x2+y2=1, and x2+y2=16. Explain in words and/or show in a picture why this would be (unnecessarily) complicated in Cartesian coordinates. Then, setup and evaluate the integral using polar coordinates.arrow_forwardEvaluate the line integral where F=√(1+x3) i + 2xy j and C is given by triangle with vertices (0, 0) (1, 0) , (1, 3) with counterclockwise orientation.arrow_forward14. Evaluate the iterated integral by converting to polar coordinates 2 2-y2 (x+y)dx dyarrow_forward
- Use Green's Theorem to evaluate the line integral along the given positively oriented curve. integral cos y dx + x2 sin y dy C is the rectangle with vertices (0, 0), (5, 0), (5, 4), (0, 4)arrow_forwardUse Green's Theorem to evaluate the line integral along the given positively oriented curve. C cos(y) dx + x2 sin(y) dy C is the rectangle with vertices (0, 0), (2, 0), (2, 1), (0, 1)arrow_forwardDefine an integral (in image)5.1 Draw a projection on the yz plane. 5.2 Convert the integral to the cylindrical coordinate system and calculate the integral.arrow_forward
- A) Evaluate the given line integral directly. B) Evaluate the given line integral by using Green's theorem.arrow_forwardEvaluate the iterated integral by converting to polar coordinates integral{integral(x^2+y^2)dy from 0 to root(2x-x^2)}dx from 0 to 2arrow_forwardHow can you change a double integral in rectangular coordi-nates into a double integral in polar coordinates? Why might it be worthwhile to do so? Give an example.arrow_forward
- I don't understand why the double integral of z dA over region D turns into [(64-4r^2)^(1/2)-(-(64-4r^2)^(1/2)]rdrdtheta, instead of just (64-4r^2)^(1/2) rdrdtheta since R {(r,theta) | 0 <= r <= 2; 0 <= theta <+ 2pi}.arrow_forwardUse Green's Theorem to evaluate the line integral along the given positively oriented curve. ∫C xy2 dx + 4x2y dyC is the triangle with vertices (0, 0), (3, 3), and (3, 6)arrow_forwardSketch the area bounded by the circle r = 5 and the rays θ = π2 and θ = π, and compute its area as an integral in polar coordinates.arrow_forward
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