#$&* course Mth 277 Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises.
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Given Solution: M is the coefficient of dx, N the coefficient of dy. So for the first integral, M(x, y) = 3 x^2 y and N(x, y) = 5 x e^sqrt(x^2 + y^2) For the second integral, N(x, y) = sqrt( x y) and M(x, y) = - x y / (x^2 + y^2). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: `q002. What is the integrand for the area integral corresponding to each of the integrals in the preceding? Those integrals are integral( 3 x^2 y dx + 5 x e^sqrt(x^2 + y^2) dy, integrated along the arc of a circle of radius 1 centered at the origin). and integral( sqrt( x y) dy - x y / (x^2 + y^2) dx, integrated around boundary of the rectangle 1 <= x <= 3, 1 <= y <= 2). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: int(3x^2ydx+5xe^sqrt(x^2+y^2)dy cent at origin,r=1 M(x,y)=3x^2y M_y = 3 x^2 M_x=6xy N(x,y)=5xe^sqrt(x^2+y^2) N_x=5e^sqrt(x^2+y^2)+10xye^sqrt(x^2+y^2) int(sqrt(xy)dy-xy/(x^2+y^2)dx 1 <= x <= 3, 1 <= y <= 2) M(x, y)=-xy/(x^2+y^2) N(x, y)=sqrt(x) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The area integral has integrand N_x - M_y . For the first integral , M(x, y) = 3 x^2 y and N(x, y) = 5 x e^sqrt(x^2 + y^2). So M_y = 3 x^2, and N_x = 5 e^sqrt(x^2 + y^2) + 10 x y e^sqrt(x^2 + y^2). The integrand is therefore 3 x^2 - (5 e^sqrt(x^2 + y^2) + 10 x y e^sqrt(x^2 + y^2)). For the second integral, N(x, y) = sqrt( x y) and M(x, y) = - x y / (x^2 + y^2). The integrand is N_x - M_y = y / (2 sqrt( x y ) ) - ( -x / (x^2 + y^2) + 2 x y^2 / (x^2 + y^2)^2). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: `q003. Parameterize the curves and set up the integrals given in the first question. The integrals are integral( 3 x^2 y dx + 5 x e^sqrt(x^2 + y^2) dy, integrated along the arc of a circle of radius 1 centered at the origin). integral( sqrt( x y) dy - x y / (x^2 + y^2) dx, integrated around boundary of the rectangle 1 <= x <= 3, 1 <= y <= 2) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: int(3x^2ydx+5xe^sqrt(x^2+y^2)dy r=1 C(0,0) int(-3cos^2(t)sin^2(t)+5cos^2(t) 0 to 2 pi int(sqrt(xy)dy-xy/(x^2+y^2)dx1<=x<=3,1<=y<=2) The rect is giving me some trouble confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The circle of the first integral can be parameterized by x(t) = cos(t), y(t) = sin(t), 0 <= t M= 2 pi. Note that this parameterization takes use in the positive (i.e., counterclockwise) direction around the circle. With this parameterization out integral is integral ( 3 cos^2(t) sin(t) * (-sin(t) dt) + 5 cos(t) e^sqrt(cos^2(t) + sin^2(t)) * cos(t) dt, t from 0 to 2 pi) = integral ( -3 cos^2(t) sin^2(t) + 5 cos^2(t), t from 0 to 2 pi). This integral is perfectly feasible using standard methods. The rectangle of the second integral can be parameterized on each of its four sides. The first side runs from x = 1 to x = 3, with y = 1, so that dy = 0, yielding integral( -x * 1 / (x^2 + 1^2) dx, x from 1 to 3) . The integrand simplifies to -x / (1 + x^2). The second side runs from y = 1 to y = 2, with x = 3, so that dx = 0, yielding integral (sqrt(3 y) dy, y from 2 to 3). The third side runs from x = 3 to x = 1 with y = 2 and dy = 0, yielding integral( -2 x / (x^2 + 4) dx, x from 3 to 1). The fourth side yields integral ( sqrt(y) dy, y from 2 to 1). All these integrals are straightforward (the first and third use substitution u = 1 + x^2 and u = 4 + x^2, respectively). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: