Get Ready: The region R is bounded by the curves y = x 2 + 1 y = x + 3. a. Find the area of region R. b. The region R is revolved around the horizontal line y = 1. Find the volume of the solid formed. c. The region R is revolved around the horizontal line y = 8. Find the volume of the solid formed. I. Finding the Volume of a Solid with a known Base The base of a solid is the shape of a region between the x- axis y = 4sin x. Each cross section cut perpendicular to the x- axis is a semicircle. Find the volume of the solid. 2. The base of a solid is the shape of a region between the x- axis y = x 2 + 4x. Each cross section cut perpendicular to the x- axis is a semicircle. Find the volume of the solid.
3. The base of a solid is the shape of a region between the x- axis y = x 2 + 4x. Each cross section cut perpendicular to the x- axis is a square. Find the volume of the solid. 4. The base of a solid is the shape of a region between the x- axis y = x 2 + 4x. Each cross section cut perpendicular to the x- axis is a rectangle with height five times the length. Find the volume of the solid. 5. The base of a solid is the shape of a region between the x- axis y = x 2 + 4x. Each cross section cut perpendicular to the x- axis is an equilateral triangle. Find the volume of the solid. 6. The base of a solid is the shape of a region between the x- axis y = x 2 + 4x. Each cross section cut perpendicular to the x- axis is an isosceles right triangle with one leg across the base of the solid. Find the volume of the solid.
II. Known Base is Area between two curves 1. Find the volume of the solid whose base is bounded by the lines y = x 4, y = 4 x, x = 0 with the indicated cross sections taken perpendicular to the x- axis: a. Squares b. Semi- Circles c. Rectangles whose height is 3 times the base d. Equilateral Triangles e. isosceles right triangle with one leg across the base of the solid
2. Find the volume of the solid whose base is bounded by the lines y = x 2 x 3 y = x with the indicated cross sections taken perpendicular to the x- axis: a. Squares b. Semi- Circles c. Rectangles whose height is 5 times the base d. Equilateral Triangles e. isosceles right triangle with one leg across the base of the solid
III. Let R be the region in the first quadrant by the graphs of y = x 2 + 1 y = 4x + 1, as shown in the figure below. a. Find the area of region R. b. Find the volume of the solid generated when R is revolved about the horizontal line y= - 2. c. Find the volume of the solid generated when R is revolved about the horizontal line y= 18. d. The region R is the base of a solid. For this solid, at each x the cross section perpendicular to the x- axis is a square. Find the volume of this region. e. In another case, the region R is the base of a solid. For this solid, the cross- section perpendicular to the x- axis is a rectangle with a height 4 times the length of its base in region R. f. In another case, the region R is the base of a solid. For this solid, the cross- section perpendicular to the x- axis is a semicircle with diameter equal to its base in region R.