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 revolution about the x-axis over the interval [0, 2]

4. A spring whose equilibrium length of 15in. exerts a force of 50 lbs when it is stretched to 20in. Find the work required to stretch the spring from 22in. to 24in.

Expert Solution

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...

(18) The region is bounded by y = 2 − x 2 and y = x....

(18) The region is bounded by y = 2 − x 2 and y = x. (a) Sketch the region. (b) Find the area of the region. (c) Use the method of cylindrical shells to set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region about the line x = −3. (d) Use the disk or washer method to set up, but do not evaluate, an integral for the volume of...

Consider a region R bounded by the y-axis, the line segment y=8-x for x from 0...

Consider a region R bounded by the y-axis, the line segment y=8-x for x from 0 to 8, and part of the circle y=-sqrt(64-x^2) for x from 0 to 8. Find the centroid.

Consider the region bounded between y = 3 + 2x - x^2 and y = e^x...

Consider the region bounded between y = 3 + 2x - x^2 and y = e^x + 2 . Include a sketch of the region (labeling key points) and use it to set up an integral that will give you the volume of the solid of revolution that is obtained by revolving the shaded region around the x-axis, using the... (a) Washer Method (b) Shell Method (c) Choose the integral that would be simplest to integrate by hand and integrate...

Consider the region R, which is bounded by the curves y=3x and x=y(4−y). (a) Set up, but...

Consider the region R, which is bounded by the curves y=3x and x=y(4−y). (a) Set up, but DO NOT SOLVE, an integral to find the area of the region RR. (b) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the xx-axis. (c) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the line x=−5x=−5.

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.

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

find the volume of the region bounded by y=sin(x) and y=x^2 revolved around y=1

Find the centroid of the region bounded by the given curves. y=x^2 , x=y^2

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.

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