Chapter 1. calc_tutorial_15_2_015

1.1 Problem Statement

{5,7,11,13}
2*$a
round($a*11/60,2)

Calculate the integral of f(x, y) = $ax over the region D bounded above by y = x(2 − x) and below by x = y(2 − y). Hint: Apply the quadratic formula to the lower boundary curve to solve for y as a function of x.

1.2 Step 1

Question Sequence

Question 1.1

The first step is to understand the region D.

The upper bound, y = x(2 − x), is a WJev8CoAMBMKOcu+7SFFhK0RgPBY9KdfgK+odVLPO9Nv/MJr84ejnQ== that opens oOGGwcKbq8CqNrJErUOSRAzZ/Vh7nAHF2SYBIUY6CY75yqquY57jULk7ItM= with x-intercepts at x = 0 and x = 2.

The lower bound, x = y(2 − y), is a WJev8CoAMBMKOcu+7SFFhK0RgPBY9KdfgK+odVLPO9Nv/MJr84ejnQ== that opens 5dS1Sso6UmgOUtHbR9/Gczrb8EQ9F7jJTfOuneJwVDkOBOFwItgd4eZCdlY= with y-intercepts at y = 0 and y = 2.

With this information, graph the region.

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Consider the following figure.

We could treat the region D as either vertically or horizontally simple, but since the integrand, x, is without the variable y, it will be easier to consider D as vertically simple.

In order to do so, we need the bounding functions in terms of x. Hence we must solve the lower bound, x = y(2 − y), for y.

Use the quadratic equation to solve for y in terms of x.

x = y(2 − y)

y2 − 2y + x = 0

y =

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

The two solutions correspond to the upper and the lower branch of the parabola. The lower bound of D is only one of these branches.

Find the proper branch that bounds D from below. (Hint: This branch goes to −infinity.)

y =

hkWPIG1V7UVWm0ta/8BFgoWV64FBI12AycuhBb2ZPHG9ulV6C65GAyggN52bBRm3HQlafyJqOVSbYXWDVb9tf2zB3TF9sWEoOqZrEr0vGAfcVhOFlg0OebwdrqH0Y289TNHTEA/sHTizJlsk5fy6nkcRu9nVlBZaw10V7RTfQ88W5qeWgXaoalBFVowq5puhFpqTXKXSA0Oxi37X7YG7E8NqEigUx6dl3kf4Dx6wEfE7D8SR6PGDPA0Gd46oNcoc70JiqJzbPhKf+S9ULNNtngI/VkGn74OMN7fzrflp4Tc0PNU5WX2QBpw2xAQngBiV2zRvRE++inpfzbs6Z0BwvPyrQmFjbmsuBjXXnxh1WrISDzCawId1F9eYQmQN7zkSJ4fMqai09jSrmKXjNPVKWlP2B3AdlBKUgMXScne5f9T2+32fg9KqUlcfVbCuQNvilCS3dncKNvGT/QSEuZhlJJl98as8rfzp4Q8Hyg==

State the inequalities that define region D

0 ≤ x ≤ 0VV1JcqyBrI=

and

1 9DxysKU6P+4= √(1-x) ≤ y ≤ x(2 − x)

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Set up the integral considering the region D as vertically simple.

,

where b =

lqzIqqZ5ggJvDKvR0tUxCGG0Olif8qwpgq+vuUFaZMpNRvHlr0cr8ddictd8uuipWhfsp+Xx7lIsAPiIt3ALAZvf/cBbEj/pYg8YDmC5LUCaxKv0Q4oqV3YdFc2RPgo772XnFDtFsvzaheIoH9udYs8GtLN/j3yxbjygBIfVlCcxb9QwsoNqPpNAGPjqUBfOgKmrBRhJr/dWF3XljRiuoFQB9Xs4gjH9QEMEXxz9ttRSeCqMODlRE5BgINfhWAmpHp+/YCIFKN4PNTn6Lhlu5zc15ceVBqBws424RTSZEF4YzzpVmfuDoez0mb+oOHBhxoukqMiNdUFTDRCFhLNoGKYdmLmSLUT2Ui3g1NQKx/pbLqa0D/i8mMS4sZsfminAhFG4KqWKGZqtEzqJp/ogbPx+wrKx4NuW3Fh6RLAUVVOjqn6DAnyvri1LAawi7jrMkYCgYhwS2a2RHNMhBTTtzJyU5/Z/yiEUUxYT6pw1CcF08Nh58q0xktJmOU97sB8RJCuBmVS5qPCccyP1DyPoXNkwopIwR/K0hpsIX0AP1n7l+tsh6uZQQl04B+/YzwJ3hzDIgOdSd7+JvNd64/z7JHhqNOM0OPJBZ4wsKWi15iPbHEjF3zMkfokehdl2mQjcHZAzEXowbW2u7Jq8YligdGxzCn5EXtr9Jvl+kDQ/XDJSpL78b4A5d/Sfl4S8TStuQGNpTejRQ9D0lqG29PvZ0e2d4wL8bU6S2tAU4aN8UVTgM5IJ2VwohMG0xNUaL7nXDgP93U96jWPvLeA1qW1pC+cTDf3KVgTMXVBjCtk4xnTCVAatF+IKjgWKUw74DaA50BycTVHFsvVdPY7L6uF3ekdpppYzhoNSPc+8L+kCnk2oZIWVJK245SFdAtw7cXJMmp1X4yI0UokVR5Gvc7XLwT61ogm7xKhIzyYM3IsT/SR9queI
Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Integrate. (Hint: For the x√(1 − x) term, integrate using u-substitution.) Round your answer to two decimal places.

= SFgqQUkJGdg=

Correct.
Incorrect.