Chapter 1. calc_tutorial_15_2_039

1.1 Problem Statement

{8,12,16,20,24}
$a/2
$b/2
round(pow($b,2)/2,2)
$c+$d

Calculate the double integral of f(x, y) over the triangle in the figure below.

1.2 Step 1

Question Sequence

Question 1.1

In order to integrate over this triangular region, we need the equations of the upper and lower bounds.

Find the equation of the lower bound.

y =

7/fFoZfX33KQSL+fvIitdeWKH1dSO8krL9SSk3Lg7K3mbO0DibNCE2Ae9FjlciYcZ8yjOPCCJ8C9KUPafJkmhU7uJ8aaIYT7z3ULV8wsCVCKGkioYO+93ZMctCSx9vfkq9E5eCUSSEM/xM0XKSSHUKCBrQJ0LddmGFUr5W7bCOIITN5HtiwNoX+ZzX3oS5o1Ij6c36Hrxf/Km+Z7F1nzDJZTaXvbw2z/K2phqRCwp7UP6mcOXG2cbFS144mC7l1qOKlLn/mxq2fQR2eRgKKUsDwz2lphRxTWVK7wUbeogdpg2H6MT7tCrzu7969z454lZ4WWC7hHLQDpVxcaXI6AC535/lDR5+PIWbUQ0iQnmhetySOAz9m93WfQDa3kd1VUYc4ejXYUyu3QN8DWO183DgOxqSf/M0Ls+wwtCOvtbWnOgGBxmwP9V5iapgFK9ipT/VmJiMx7NI9YjwVSW9IvLI8NNFHarm6Epl52q09qYp7DjmgOpSkjHebzNkudCxywLum8xE8/9wCo6cyLApFvqHRWRiXCOnF8CdoAC+H+CnyA62EyPtGeUMTncQLADmPfNnhx/9Ta8QmPV8xKbTvmVygNr6zTfyZksm7/gL6cZ0a0VkXJu3Asa7SW2NMVdz2o514BMiqknsXW9MSxwyPyxYxYQxu1032aDDPljKxOQzkV9HsuP0tlu0ppMttPBJeUsUtv+XO3xJhSyOVICrFSr3mznbEQZbSOIVEoXTizGBrT6ByuGzTJC2A5/32WCxOk7RLvEwdPHtvNs6xbHv1s4Al3hzalIwIC/gDJY5RdskfW4ks61jUqCHszHi2OugNmu/c5bi2KAXP02lXWtiEJnRrExawQ2dTNX+1kXc5PcBnaWKiwhN/e3NlA9zmHayiGVk7iXvbI7Tw6mJFT0Je24nsSjrahNOM2+TN8AYBjgCkc2UtI

Find the equation of the upper bound.

y =

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

Since these bounds are in terms of x and the integrand is without the variable y, it is easiest to express the triangle as M2jxtJGVDpVt/Kx8Ux1JJ4ofYkdaWFI9ZSKzvQ== simple.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Consider the following figure.

State the inequalities of the vertically simple region defined by the triangle.

0 ≤ xh4XZagboIgc=

and

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

Set up the double integral with the triangle as vertically simple.

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
Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Integrate.

A/k0sZwcoJ5gHWHY4ehtaWsdSbwym9U34ISrp2NacNTHqqEhiHU0+rS/xOSE5IPGqAbvAgIof2c+9C8XthwE9iUzC1ylFcC7LrlQS6u21QgAx8R7Nwmg8uKgmErYvZD/ykICxxZrPrMunCu8hpOWCzojZR+RKo3Ybk4LAJ216uWzA/yRajg6ffoGOYWLCYFM2ApizuxMkJ0lxXmMsl/3lOd4MBGHDWN/mHhTm8RqpMF+2J+B5kHhMiNd+ROZnS/ALCdE8m4+jCa/uRVSUWket+PyAQuOOoETCy6N8tyv2jPZhFqID7VuiTbE9nWPMF0/KbGYDDPFQeyBiXkcsjPwj+xJfMVsVFzwJqkvrW0MvWh0uRY6S840vBdWhfG62pXiifgN70SZZovMBVbDdsY1Ht5Gyllr0at9JKq5o7c5dD4C7gUqsDcy5Hg3/Z8ICpZjyoGaC8j4UQsT5zNguqvCt2XccV/0n0bXWyF6/DYl/plaFBfpn93lQSN+/HrpHXRgO+UCgqD3RvFKDkpiR6kWYA8kebwuMtUMc5ZCpSe0MHJqnHSuEFsr/sIQ51dc//meVT/zcD/qcKJw9V5ed1aRjdENgf3gYn5EVktA0K8ysoBuzrqurC8JgeUWkgjclOp79ufmm6pyRajnktyEEUNxRrwHFF2SnrqPKJoebJ5kLe8/UcYf1ADTPLy6dQI/D6pqnYphoaKee+kR1uFkhNmIEzjckxLKW1zXIgQTeee5I52BFslFnYxm/IqoaeQ9IVCWje4HYLlLNJUmvdv1NeYDJkMNJBfX7ZA51wG8GaClM67TCoYEk1IDYbJhWowXw/a1MVE5Huq+8W/IQRAAOCh86N1P4o+osRe937031d34EAB3ja1Fu1EcvgwztCLFURoGhoqvSnSZVm286Ry+y8pcA8G+kG37PAK+LQcX6vAXo87Y1a3rqJjFwo9/ibZaqKosNsaYTVq3rmdaBEgzyeP2Pu5PSqnzcXf474vufUa0gh4KcoxXPsa7hDI7UNGd3LOqY+8PQ2/w9l2GzeXWkWRu/NOP+8nBR58r3JyvBg9pKsOMgAyJTLCqonbeOLGjtS4ItDzf4ptten0VsJzBCPZ7ZTZxFw/4MAIitTcPGznEMHnk12KixUYLpLksjgrg9vywrGVFGCuZ32wlO7CLA33IRb0O40ADsI4HfS8z0ejNgIp8OVzoYJz55r6/LMVRqZSO3/mtW1z62vxdjYk3lCEA0hedR/MLc1g5unBMUzXo1tr44RAv0s76ruUHV71a0UTpZE0WX5jGMKMT5OGG/PL3FiaDh7eQLM7HGZnk7wKoknkuQXaQ

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
Correct.
Incorrect.