Finding Surface Area In Exercises 1–14, find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the
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Calculus
- ulus III |Uni Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively oriented curve C, where C is the rectangle with vertices(0,0), (4, 0), (4, 2) and (0, 2).arrow_forwardDetermine the distance y measured from the x axis to the centroid of the area of the triangle shown in Fig. 9-10. (b - x), (x.y) (1,P) Fig. 9-10arrow_forwardAn integral expression for the area of the surface generated by rotating the line segment joining (3, 1) and (6, 2) about the x-axis is ______.arrow_forward
- -V2/2 4-x I 2 Vx2 + y² + 3 dy dx + 2 Vx2 + y² + 3 dy dx -V2 V2/2 J VI-x² Rewrite as an iterated double integral in polar coordinates and evaluate.arrow_forwardQuestion 6 Use Green's Theorem to evaluate F dr. where F(2, y) = (9xy, y" -8) and C is the rectangle with vertices (2, 2), (5,-2), (5,1), and (2,1). The integrat obtained from from Green's Theorem is dA where D is the interier of the rectangle. This evaluates to Question Help: Message instructor Check Answer primearrow_forward"[₁ fx, y) de by dividing the rectangle R with vertices (0, 0), (4,0), (4, 2), and (0, 2) into eight equal squares and finding the sum Approximate the integral center of the th square Evaluate the iterated integral and compare it with the approximation Step 1 The vertices of the rectangle are given as (0, 0), (4, 0), (4,2), and (0,2) As the x-coordinate varies from 0 to 4, the length of the rectangle is Therefore, the area of the rectangle is Sum SAUDIYou cannot come back) As the y coordinate varies from 1 to 2, the breadth of the rectangle is Σ, where is thearrow_forward
- Using polar coordinates to evaluate (3x + 4y) dA, where R is the region in the upper half of the washer bounded by x² + y? = 1 and x² + y? = 4. 2.arrow_forwardArea of Plane Region 3. R: x2 + 3y = 4 and x − 2y = 4.4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forwardIT (z+) dz dy. Your answer must include a 4a.) Set up the Integral in polar coordinates: sketch of the region to help explain your steps. (10] 4b.) Evaluate the integral I (z+y) dz dy. You can either do it directly OR using your equiv- alent expression In polar form. (Do NOT do it both ways.) [10)arrow_forward
- c3 9-a2 J. 3. Sketch the region, convert the integral (x²+y²)²dydx into polar coordinates and evaluate. V9-x² y 2 1 -2 -1 -1+ -3 3 -2 -3arrow_forwardHow do you find the area of a region 0 ≤ r1(θ) ≤ r ≤ r2(θ),a≤ θ ≤ b, in the polar coordinate plane? Give examples.arrow_forwardCalculus Evaluate the following integrals by changing to polar coordinates: Part A) Let R be the region in the first quadrant enclosed by the circle x^2 + y^2 = 16 and the lines x = 0 and y = x. [[ cos(x² + y²)dA Rarrow_forward
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