Volume of a solid

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

(a) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{cc} x=19 \sqrt{2 y}, & x=4 \sqrt{2 y} \\ z=0, & z+y=2 \end{array} .\right. \]

9.5.1 Volume of a solid

1.21 $

3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{c} x^{2}+y^{2}=3 y \\ x^{2}+y^{2}=6 y \\ z=0, \quad z=\sqrt{x^{2}+y^{2}} \end{array}\right. \]

9.5.2 Volume of a solid

1.45 $

(3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{c} x=y^{2}-2, \quad x=-4 y^{2}+3 \\ z=\sqrt{16-x^{2}-y^{2}}+2, \quad z=\sqrt{16-x^{2}-y^{2}}-1 \end{array}\right. \]

9.5.3 Volume of a solid

1.21 $

3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V: z=30\left((x+1)^{2}+y^{2}\right)+1, \quad z=60 x+61 . \]

9.5.5 Volume of a solid

1.45 $

3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ z=\sqrt{8-x^{2}-y^{2}} \\ z=0 \end{array}\right. \]

9.5.11 Volume of a solid

1.45 $

(3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} y=\sqrt{x^{2}+z^{2}} \\ y=2-x^{2}-z^{2} \end{array}\right. \]

9.5.12 Volume of a solid

1.45 $

3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} z=6 \sqrt{x^{2}+y^{2}} \\ z=16-x^{2}-y^{2} \end{array} .\right. \]

9.5.4 Volume of a solid

1.45 $

3 Problem: Calculate the volume of the solid: \[ V:\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}} \leq \ln \frac{x^{2}+y^{2}+z^{2}}{x^{2}+y^{2}} . \]

9.5.8 Volume of a solid

3.13 $

30 Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} y=\sqrt{x^{2}+z^{2}} \\ z=2-x^{2}-z^{2} \end{array} .\right. \]

9.5.13 Volume of a solid

1.21 $

(a) Problem: Calculate the volume of the solid bounded by the given surfaces using a triple integral. Plot the drawings of the given solid and its projection on plane \( X O Y \). \[ x^{2}+y^{2}=2 y, \quad z=\frac{13}{4}-x^{2}, \quad z=0 \]

9.5.17 Volume of a solid

1.45 $

a) Problem: Find the volume of a cylindrical solid bounded by surfaces \( z=16-y^{2}, x^{2}+y^{2}=2 x, z=0 \).

9.5.18 Volume of a solid

0.96 $

(3) Problem: Calculate the volume of a solid bounded by the given surfaces using a triple integral. Plot drawings of this solid and its projection on plane XOY. \[ z=0, \quad z-4 \sqrt{y}=0, \quad x=0, \quad x+y=4 \]

9.5.19 Volume of a solid

1.21 $

(a) Problem: Calculate the volume of the solid bounded by the surfaces: \[ z=\sqrt{x^{2}+y^{2}}, \quad z=1-\sqrt{1-x^{2}-y^{2}} . \]

9.5.14 Volume of a solid

1.69 $

3) Problem: Calculate the volume of a solid bounded by the given surfaces using a triple integral. Plot the drawings of the given solid and its projection on XOY. \[ z=0, \quad z=4-x-y, \quad x^{2}+y^{2}=4 \]

9.5.15 Volume of a solid

2.89 $

Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} x=16 \sqrt{2 y}, \quad x=\sqrt{2 y} \\ z+y=2, \quad z=0 \end{array}\right. \]

9.5.6 Volume of a solid

1.45 $

1
2
›