please show all your work :)

Using concepts from the solids of revolution, write out (but do not evaluate!) an integral that computes the volume of a torus with inner radius R = 5 and outer radius = 2, by completing the following steps: i. Identify the region and the axis of revolution. Provide a graph of the region (demos.com might help with this!). ii. Choose your desired slicing method and write out the corresponding area function. iii. Determine the bounds of integration. iv. Write the integral(s) that will compute the volume of the solid (you do not need to evaluate it!). Left: The full solid volume. Right: solid with a cutout so that the radii can be identified. Step 3: Determine the Buds of Integration The circle we are revolving extends from OC = 3 to DC = 7 Why ? The center of the circle is at x = 5 and the radius is 2 So the left edge of the circle is at 0c= 5-2=3 and the right edge is at oc = 5+ 2= 7 Thus the bounds of integration for a are from 20= 3 to ze = 7 Step 4: Set up the integral Height of the shell ." The height of the shell at each 2 is the vertical distance between the top and bottom of the circle at that value from the equation of the circle, the height is given by height of shell = 2* \\ +2 - (x - R ) " = 2,4-(x-5) This is the distance between the upper half and lower half of the circle Shell volume ." the volume of the tones is obtained by revolving this legion around the y-axisusing the shell method. V= 1 25LI. (2)4-(2c-5)2) doc Simplyying ! V= 45(20 4- ( 20- 5) 2 docStep 2: Choose the slicing Method (shell Method) We will use the shell method for finding the volume of the solid of revolution . The shell method is ideal here because the region being evolved is described as a function of sc . and we are revolving around a vertical axis Formula for the shell Method When revolving the y-axis, the volume of a aplindrica shellic given by V = / Role . height of shell doc Where 20 is the distance from the axis of revolution ( which is the yaxis ) The height of the shell is the function that describes the tap and bottom of the region being revolved.Step li Identify the Region and Axis of Revolution Region The region to be revolved is a circle with radius r= 2 , centered at the point for ( 5, 0) Equation of the circle ( - 5 )+42 = 4 Axis of Revolution The otis of revolution is the y-axis ( the vertical asis at se=0 ) this means the arcle will be revolved around the 4-axis to form the 30 shape of the torus .\f