Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis.y = 4e−x2,      y = 0,      x = 0,      x = 1Sketch the region and a typical shell.Step 1Rotating a vertical strip around the y-axis creates a cylinder with radius r = $$x and height h = $$4e−x2 . Sketch the region and a typical shell. Step 2Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 2𝜋x  dx.

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.xy = 2,    x = 0,    y = 2,    y = 4

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.x = 4 + (y − 5)2,  x = 20V = 2194.2

Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 2𝜋x

Step 2Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 1 1 2𝜋x $$4e−x2  dx.0Step 3The integral 2𝜋 4xe−x2 dx can be done with the substitution u = $$−x2 and du = $$−2x dx.Step 4With the substitution, we have 2𝜋 4xe−x2 dx = − $$4π eu du = − $$4πeu + C.Step 5Going back to x, the volume of our solid is

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.y = x3/2,    y = 8,    x = 0