(b) Set up (but do not evaluate) the integral in spherical coordinates that gives the volume of E. Make sure to show ALL your work about how you found the bounds for 8. & and 8.
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- 2. Chapter 15 Review 33: Use polar coordinates to calculate sD√x2 + y2dA where D is the region inthe first quadrant bounded by the spiral r = θ, the circle r = 1, and the x-axis.10.2 37)Find the area enclosed by the given parametric curve and the y-axis.Consider the surface S shown in the graph, whose parametrization is given by: r (u, v) = (1 + 2v, 3uv, 4 - u2), where 0 ≤ u ≤ 2, 0 ≤ v ≤ 2 (see attached image with the graph) The surface differential, dS, is given by: (see image with possible answers)
- Find the area enclosed by the simple closed curve given by the parametric equations, x = 2cost + cos2t and y = 2sint − sin2t for t ∈ [0,2π]. The graph is shown below.Answer question 4 4. Region bounded by y = x 2, x=0 and y=15, rotated about the y-axisThe diagram shows a conical section taken out of a sphere. Use spherical polar coordinates to evaluate the volume of this shape in terms of R and α. Verify your answer is correct for the cases α = 0, α = π.
- Using the concept of triple integrals in spherical coordinates ( theta , rho and r ) , solve the problem by drawing a graph .Question 7. Consider the solid in xyz-space, which contains all points (x, y, z) whose z-coordinate satisfies 0 ≤ z ≤ 4 − x2 − y2 . Which statements do hold? a) The solid is a sphere. b) The solid is a pyramid. c) Its volume is 8π. d) Its volume is 16π . 3Question 8: Find the volume of the solid bounded above by the surface z = f(x, y) = 2x+y and below by the plane region R. R is a triangle bounded by y = 2x, y = 0, and x = 2
- Use double integral to find the volume of the cylinder-like object that exists above the first quadrant on the xy plane and below the plane z = 10. The cross section of this "cylinder" on the xy plane is given by the polar equation r = 50 sin (2θ)Evaluate ∫ ∫ √ (x2+ y2) dA where R is the portion of the annulus 1 ≤ x2 + y2 ≤ 16 with y ≤ 0 using polar coordinates.PART 1: DETERMINE THE CENTROID IN X (? ") and THE MOMENT INERTIA IN (Iy) OF THE FOLLOWING FIGURE