 Hello, and welcome. In this screencast, we will look at an example of calculating the volume of a solid that is created by revolving a two-dimensional region around the x-axis. So we'll find the volume of the solid that's generated by revolving this two-dimensional region that's bounded by the graph of y equals x squared plus 2, this curve, which intersects the y-axis at y equals 2, by the line y equals 7, and also by the x-axis itself. So we see this two-dimensional region that is completely enclosed by those three lines. When we rotate this two-dimensional region around the y-axis, the result is this bowl-shaped three-dimensional object. Well, we don't have a formula for calculating the volume of a bowl-shaped object, so instead we're going to approximate this volume by cutting it into nicely shaped slices, and then we can improve on that approximation and find the exact volume of our object. So first, we need to decide how to slice our object to help us approximate its volume using a nice shape. We're going to slice this object horizontally because we get nice round cross sections, and we'll draw a typical slice here on our graph. Since we're slicing this horizontally, that means we're going to integrate with respect to the variable y instead of integrating with respect to x like we normally do when we make vertical slices. We're going to approximate each slice with a circular disc, and we start slicing up our object at the bottom where y is equal to 2, and slice it up until the top of the object where y is equal to 7. So those will be our limits of integration at y equals 2 and y equals 7. The thickness of each slice is going to be delta y. Next, we need to find the radius of our circular discs. So we need the distance from the center of the slice out to the parabola that forms the outer boundary of our slice. So that is a horizontal distance, so we're going to solve the equation y equals x squared plus 2 for the variable x. To do this, we subtract 2 from both sides of the equation, then we take the square root of both sides, and so our result is the radius of our slice, the square root of y minus 2. Each slice is a circular disc, and the general formula for the volume of this disc is pi r squared h, where r is the radius of the slice and h is the thickness of the slice. When we substitute the square root of y minus 2 for the radius and delta y for the thickness of the slice, the result is pi times the square root of y minus 2 squared times delta y, which we can then simplify to pi times y minus 2 delta y. When we add up the volume of the slices, we get an approximation of the volume of the solid. We can improve this approximation by using more and more slices, and when we take the limit as the number of slices approaches infinity, we get a definite integral that will calculate the exact volume of our object. So here we see our definite integral pi times the integral from 2 to 7 of y minus 2 dy. To evaluate this definite integral, we see that we factored out the pi, so we need to find the antiderivative of y minus 2, which is 1 half y squared minus 2 y. Then we evaluate this antiderivative at y equals 7 and y equals 2 and do the appropriate subtraction and multiply the result by pi. When we're done with our calculations and simplified our result, our final result is 12.5 pi, which is the exact volume of our solid of revolution that we created by rotating a two dimensional region around the y-axis. Thanks for watching.