Compute the area of the region in the ry-plane wich is bounded by the z-axis and the curve r(t) = (t - sin(t), t²(1-t)) where 0≤t≤2m. (Kap. 16,3)
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- Find the area bounded by the following: 1. Curve y²-8y-8x +32= 0 and y²-8y + 4x = 07.7 11) Use (a) the Trapezoidal Rule and (b) the Midpoint Rule to approximate the given integral with the specified value of n.Let D be the region bounded by the parabola y = x2 and the curvey = sin x, and let P represent a path going around D counterclockwise.Compute ∫P F ·dr where F(x,y) = ∇f and f(x,y) = x2ye4x−y^2.
- Compute the integral ∮C [(cos x − 3y) dx + (2x − sin y) dy],where C is the closed curve that travels on the line segments from(0, 0) to (4, 0), from (4, 0) to (2, 1), and from (2, 1) to (0, 0).Suppose that a point (X,Y) is chosen at random from the region S on the xy-plane containing all points (x,y) such that x>= 0; y>= 0, and 6y+x<= 3. (a) Determine the joint p.d.f. of X and Y: (b) What is P(Y=X)? (c) What is P(X>=1)?Calculate the are bounded by the curves: B. y = ln x and x - y - 4 = 0
- Show that if φ is continuously differentiable in a given region V and on itsboundary S, then∫S φ dS =∫V ∇φ dV1) Evaluate the line integral ∫_c (2x − y) dx − (x+ 3y) dy , where C is a straight line from (1,1) to (3,5), followed by a horizontal line from (3,5) to (5,5).. 3. Compute E y 2 z 2 dV , where E is the region bounded by x 1 y 2 z 2 and x