Aim: How do we find the volume of a figure with a given base? Get Ready: The region R is bounded by the curves. y = x 2 + 1
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1 Get Ready: The region R is bounded by the curves y = x 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.
2 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.
3 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
4 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
5 III. Let R be the region in the first quadrant by the graphs of y = x 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.
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