Let R be the region inside x ^2 /9 + y^2 /25 = 1, with x...

Let R be the region inside x ^2 /9 + y^2 /25 = 1, with x ≥ 0. (That is, R is the right-hand side of the ellipse below.) (a) (15 pts) Use the change of variables x = 3u, y = 5v to transform ∫ ∫ R x dxdy to a polar coordinate integral. 3 −5 5 R (b) (5 pts) Evaluate the polar coordinate integral

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milcah answered 2 years ago

let R be a region bounded by x = 0 and x =1 and y =...

let R be a region bounded by x = 0 and x =1 and y = 0 and y = 1. Suppose the density is given by 1/y+1.Notice that R is denser near the x axis. As a result we might expect the centre of mass to be below the geometric center(1/2,1/2). Also since the density does not depend on x we do expect moment of inertia about the x axis to be 1/2. verify the moment of inertia about...

Let R be the region enclosed by the x-axis, the y-axis, the line x = 2...

Let R be the region enclosed by the x-axis, the y-axis, the line x = 2 , and the curve ? = 2?? +3? (1) Find the area of R by setting up and evaluating the integral. (2) Write, but do not evaluate, the volume of the solid generated by revolving R around the y-axis (3) Write, but do not evaluate the volume of the solid generated by revolving R around the x-axis (4) Write, but do not evaluate the...

Let R be the region bounded by the following graphs. Find the quantities below.y= (x−1)^2+ 1,y=...

Let R be the region bounded by the following graphs. Find the quantities below.y= (x−1)^2+ 1,y= 1,x= 0 a. Volume of the solid formed by rotating R about the x-axis, using the shell method. b. Volume of the solid formed by rotating R about they-axis, using the disc method.

Let R be the region that is inside the circle r = 3 sin θ and...

Let R be the region that is inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ. Find the circumference of the region R.

Consider a region R bound by the coordinate axes and y = ( 9 + x...

Consider a region R bound by the coordinate axes and y = ( 9 + x 2 ) − 1 2 on 0 ≤ x ≤ 4. a. Find the area of R. b. Suppose R is revolved about the x-axis to form a solid. Find the volume of the solid. c. Suppose R is revolved about the y-axis to form a solid. Find the volume of the solid.

Consider the region R enclosed between the curves y = 2 /x and y = 1,...

Consider the region R enclosed between the curves y = 2 /x and y = 1, between x = 1 and x = 2. Calculate the volume of the solid obtained by revolving R about the x-axis, (a) using cylindrical shells; (b) using washers

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

1. Consider the region bounded by the graph of y^2 = r^2 −x^2 (a) When this...

1. Consider the region bounded by the graph of y^2 = r^2 −x^2 (a) When this region is rotated about the x-axis a sphere of radius r is generated. Use integration to find its volume V (b) Use integration to find the surface area of such a sphere 2. Find the arc length of the curve y = 1 3 x 3/2 on [0, 60] ( 3. Consider the graph of y = x^3 . Compute the surface area of...

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And...

9) R is the region bounded by the curves ? = x^3 , y=2x+4 , And the y-axis. a) Find the area of the region. b) Set up the integral you would use to find the volume of a solid that has R as the base and square cross sections perpendicular to the x-axis.

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the...

A lamina occupies the region inside the circle x^2 + y^2 = 6x, but outside the circle x^2 + y^2 =9. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.

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