In this section we will discuss about application of integral to compute the volume of solids. Just as area is the numerical measure of a two-dimensional region, volume is the numerical measure of a three-dimensional solid.

Most of us have computed volumes of solids by using basic geometric formulas. The volume of a rectangular solid, for example, can be computed by multiplying length, width, and height: V=lbh. The formulas for the volumes of a sphere is \(V=\frac{4}{3} \pi r^3\), a cone \(V=\frac{1}{3} \pi r^2 h\), a pyramid \(V=\frac{1}{3} A h \) have also been introduced. Although some of these formulas were derived using geometry alone, all these formulas can be obtained by using integration.

Three are four common approach of computing volume of objects using integral, these are calculating volumes of solids using methods of

- cross-section (slicing method)
- disk
- washers and
- shells

#### cross-section (the slicing method)

#### disk method

#### washer method

#### shell method

#### Exercise

- Find the volume of the solid formed by rotating the region bounded by \(y=0 , y=\frac{1}{1+x^2},x=0 , x=1\) about the y -axis.
- Find the volume of the solid formed by rotating the triangular region determined by the points (0,1) , (1,1) and (1,3) about the line x=3
- Find the volume of the solid formed by rotating the triangular region determined by the points (0,1) , (1,1) and (1,3) about x-axis
- Find the volume of the solid formed by revolving the region bounded by y=sinx and the x-axis from x=0 to x=π about the y -axis.
- Find the volume of solid bounded by \(x=y^3,x=8\) and the x-axis and rotating the region about x-axis.
- Find the volume of solid bounded by \(f(x)=x+1,g(x)=(x-1)^2\) and the x-axis and rotating the region about y-axis.
- Find the volume of solid bounded by \(f(x\sqrt{x-1}+2\) rotating the region about y-axis between the lines x=1 and x=5.
- Suppose the area under \(y=-x^2+1\) is rotated around the x-axis. Find the volume of the solid of rotation.
- Suppose the area below the curve \(f(x)=x+1, x \in [0,3]\), is rotated about the y-axis. Find the volume
- Let R be the region of the xy-plane bounded below by the curve \(y=\sqrt{x}\) and, above by the line y=3. Find the volume of the solid obtained by rotating R around
- x-axis
- y-axis
- line y=3
- line x=9

- Let R be the region of the xy-plane bounded above by the curve \(x^3y=64\) and, below by the line y=1, on the left by the line x=2, and on the right by the line x=4. Find the volume of the solid obtained by rotating R around
- x-axis
- y-axis
- line y=1
- line x=2

- Use Shell Method to find the volumes of the solids generated by revolving the shaded region about the indicated axis.
- \(y=1+x^2\) about y-axis
- \(y=2-x^2\) about y-axis
- \(x=1+y^2\) about x-axis
- \(x=4-y^2\) about x-axis

- Use Shell Method to find the volumes of the solids generated by revolving the given bounded regions about the
y -axis.
- \(y=2x,y=-x,x=1\)
- \(y=x^2,y=x,y \ge 0\)
- \(y=x^2,y=x+2,x \ge 0\)
- \(y=1-x^2,y=x^2,x \ge 0\)
- \(y=\sqrt{x},y=2-x,x = 0\)
- \(y=\frac{1}{x},x=2,x=3,y=0\)

- Use Shell Method to find the volumes of the solids generated by revolving the given bounded regions about the
x -axis.
- \(x=2 \sqrt{y}, x=-y,y=4\)
- \(x=y^2, x=y, y \ge 0\)
- \(x=y-\frac{y^2}{4}, x=0\)
- \(x=y-\frac{y^2}{4}, x=\frac{y}{2}\)
- \(y=|x|,y=2\)
- \(y=x+1,y=2x,y=2\)

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