\nA solid body is defined by
\n\\(
\n0\\leq x\\leq 2,\\qquad
\n0\\leq y\\leq 3,\\qquad
\n0\\leq z\\leq x+y.
\n\\)
\n
![](\"http:\/\/www.max-sperling.bplaced.net\/wp-content\/uploads\/2026\/09\/Exercise_10.png\")
<\/p>\nPython script \u25bc<\/summary>\n\r\nimport numpy as np\r\nimport plotly.graph_objects as go\r\n\r\nfig = go.Figure()\r\n\r\n# --------------------------------------------------\r\n# 1. Bottom surface: z = 0\r\n# --------------------------------------------------\r\n\r\nx = np.linspace(0, 2, 20)\r\ny = np.linspace(0, 3, 20)\r\n\r\nX, Y = np.meshgrid(x, y)\r\n\r\nfig.add_trace(go.Surface(\r\n x=X,\r\n y=Y,\r\n z=np.zeros_like(X),\r\n opacity=0.5,\r\n showscale=False,\r\n name="z = 0"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# 2. Top surface: z = x + y\r\n# --------------------------------------------------\r\n\r\nZ = X + Y\r\n\r\nfig.add_trace(go.Surface(\r\n x=X,\r\n y=Y,\r\n z=Z,\r\n opacity=0.8,\r\n showscale=False,\r\n name="z = x + y"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# 3. Side surface: x = 0\r\n# 0 <= y <= 3\r\n# 0 <= z <= y\r\n# --------------------------------------------------\r\n\r\ny = np.linspace(0, 3, 20)\r\nz = np.linspace(0, 3, 20)\r\n\r\nY, Z = np.meshgrid(y, z)\r\n\r\n# Keep only the region below z = y\r\nZ = np.minimum(Z, Y)\r\n\r\nfig.add_trace(go.Surface(\r\n x=np.zeros_like(Y),\r\n y=Y,\r\n z=Z,\r\n opacity=0.6,\r\n showscale=False,\r\n name="x = 0"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# 4. Side surface: x = 2\r\n# 0 <= y <= 3\r\n# 0 <= z <= 2 + y\r\n# --------------------------------------------------\r\n\r\ny = np.linspace(0, 3, 20)\r\nz = np.linspace(0, 5, 20)\r\n\r\nY, Z = np.meshgrid(y, z)\r\n\r\n# Keep only the region below z = 2 + y\r\nZ = np.minimum(Z, 2 + Y)\r\n\r\nfig.add_trace(go.Surface(\r\n x=2 * np.ones_like(Y),\r\n y=Y,\r\n z=Z,\r\n opacity=0.6,\r\n showscale=False,\r\n name="x = 2"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# 5. Side surface: y = 0\r\n# 0 <= x <= 2\r\n# 0 <= z <= x\r\n# --------------------------------------------------\r\n\r\nx = np.linspace(0, 2, 20)\r\nz = np.linspace(0, 2, 20)\r\n\r\nX, Z = np.meshgrid(x, z)\r\n\r\n# Keep only the region below z = x\r\nZ = np.minimum(Z, X)\r\n\r\nfig.add_trace(go.Surface(\r\n x=X,\r\n y=np.zeros_like(X),\r\n z=Z,\r\n opacity=0.6,\r\n showscale=False,\r\n name="y = 0"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# 6. Side surface: y = 3\r\n# 0 <= x <= 2\r\n# 0 <= z <= x + 3\r\n# --------------------------------------------------\r\n\r\nx = np.linspace(0, 2, 20)\r\nz = np.linspace(0, 5, 20)\r\n\r\nX, Z = np.meshgrid(x, z)\r\n\r\n# Keep only the region below z = x + 3\r\nZ = np.minimum(Z, X + 3)\r\n\r\nfig.add_trace(go.Surface(\r\n x=X,\r\n y=3 * np.ones_like(X),\r\n z=Z,\r\n opacity=0.6,\r\n showscale=False,\r\n name="y = 3"\r\n))\r\n\r\n\r\n# --------------------------------------------------\r\n# Layout\r\n# --------------------------------------------------\r\n\r\nfig.update_layout(\r\n title="Solid: 0 \u2264 x \u2264 2, 0 \u2264 y \u2264 3, 0 \u2264 z \u2264 x + y",\r\n\r\n scene=dict(\r\n xaxis_title="x",\r\n yaxis_title="y",\r\n zaxis_title="z",\r\n aspectmode="data"\r\n )\r\n)\r\n\r\nfig.show()\r\n<\/pre>\n<\/details>\n<\/center><\/p>\n
\nFind<\/strong><\/p>\n
\nThe volume of the body.
\n<\/center><\/p>\n
\nSolution<\/strong><\/p>\n
\n\\(
\n\\begin{array}{l}
\n \\text{The volume gets calculated via:} \\\\
\n V=\\int\\int\\int_V 1\\;dV=\\int_0^2 \\int_0^3 \\int_0^{x+y} 1\\;dz\\;dy\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to z:} \\\\
\n V=\\int_0^2 \\int_0^3 [z]_0^{x+y}\\;dy\\;dx =\\int_0^2 \\int_0^3 (x+y)\\;dy\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to y:} \\\\
\n V=\\int_0^2 [xy+\\frac{y^2}{2}]_0^3\\;dx =\\int_0^2 (3x+\\frac{9}{2})\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to x:} \\\\
\n V=[\\frac{3x^2}{2}+\\frac{9x}{2}]_0^2 = 6+9 = 15. \\\\
\n\\end{array}
\n\\)
\n<\/center><\/p>\n
\nAnswer<\/strong><\/p>\n
\nThe volume is \\(15\\;m^3\\).
\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"Topics: Volume integral Given A solid body is defined by Python script \u25bc Find The volume of the body. Solution<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[47],"tags":[],"_links":{"self":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778"}],"collection":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=19778"}],"version-history":[{"count":48,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778\/revisions"}],"predecessor-version":[{"id":19886,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778\/revisions\/19886"}],"wp:attachment":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=19778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=19778"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=19778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
<\/center><\/p>\n
\n
Find<\/strong><\/p>\n Solution<\/strong><\/p>\n Answer<\/strong><\/p>\n Topics: Volume integral Given A solid body is defined by Python script \u25bc Find The volume of the body. Solution<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[47],"tags":[],"_links":{"self":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778"}],"collection":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=19778"}],"version-history":[{"count":48,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778\/revisions"}],"predecessor-version":[{"id":19886,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/19778\/revisions\/19886"}],"wp:attachment":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=19778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=19778"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=19778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nThe volume of the body.
\n<\/center><\/p>\n
\n
\n\\(
\n\\begin{array}{l}
\n \\text{The volume gets calculated via:} \\\\
\n V=\\int\\int\\int_V 1\\;dV=\\int_0^2 \\int_0^3 \\int_0^{x+y} 1\\;dz\\;dy\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to z:} \\\\
\n V=\\int_0^2 \\int_0^3 [z]_0^{x+y}\\;dy\\;dx =\\int_0^2 \\int_0^3 (x+y)\\;dy\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to y:} \\\\
\n V=\\int_0^2 [xy+\\frac{y^2}{2}]_0^3\\;dx =\\int_0^2 (3x+\\frac{9}{2})\\;dx. \\\\
\n \\\\
\n \\text{Integration with respect to x:} \\\\
\n V=[\\frac{3x^2}{2}+\\frac{9x}{2}]_0^2 = 6+9 = 15. \\\\
\n\\end{array}
\n\\)
\n<\/center><\/p>\n
\n
\nThe volume is \\(15\\;m^3\\).
\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"