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

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  dx.0

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.

Determine the volume of the solid obtained by rotating the region bounded by𝑦 = 𝑥2 − 4x + 5, x = 1, x = 4 and the x-axis about the x-axis. (78𝜋5

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

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.y = ln(x),   y = 0,   x = 4;   about the x-axis