Chapter 1. calc_tutorial_16_2_015

1.1 Problem Statement

{2,3,4,5,6,7,8}
32*$a
2*$a
2*$c
32*pow($a,2)
pow($d,2)
pow($a,2)
4+$g
$f+$g

Compute for the curve specified.

f(x, y, z) = 2x2 + $bz, c(t) = (2et, $ct2, $at), 0 ≤ t ≤ 1

1.2 Step 1

Question Sequence

Question 1.1

Recall how to compute a scalar line integral.

Let c(t) be a parametrization of a curve C for a ≤ t ≤ b. If f(x, y, z) and c'(t) are continuous, then the scalar line integral can be calculated as follows.

Thus we will need to calculate ||c'(t)||.

Find c'(t) and ||c'(t)||.

c'(t) =

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

||c'(t)|| =

CphupNFafQUv5Ej/drr1bXj1lwcrbzG+CgBwX+XEsOgOMMlg+HaHjM3q2djlzjVaMPuy3YKJmApJh/2ef3IvplugX5QIRLrOj8BcMtLTOMn++tFehg/sRJILUU2HU5Y/bG2tvVEqoy8Ag2sYHmJvMzKTRhOhg8BkaehOI9LYyiVIsYPneKErc6l5Xfl86bjklJqGTDcy8XiSQIv2r98vHUXAMCntUnE/5Xh2pyJkjjs3Zw7Iiqv7kbAVQffJMvkPGk3mIkd8C+Esac78iRapDU93ZDaJRBYCOUQi6A8i6835Vf8q0hfALeTEqPwSHVH9FeBQ8mvGqy47vGhxm/6+sUEgaCuLaO8d5fcIPvtkXYCQkjc0ZG5WHISXwRD/7GeZOJeULudoHN1n6MOegT02pZQytpOJ9hBf9WoqADWqmGl0ejOVkyhC45wZHqtbK58YAqJDJ0HFisHm92bxSCm1qUABM2/PmjSRXnEeI8RH2PGVvfvJexLF107+lsjF2HMs4Rvszg3/lyGgA//lcx6V/CK+5ZAnXvJsY4UPx2gC3jV6SOzdOWwEi1C/P0wZ/yy0oAWizTUsEm7W2sps8g1JKkwEWcp2afbVVVdGAAN83/bkb8vjslXV2mxNGovTDbS9W3QRSMN2XIAKShWBi9jHUOQsIHNs9BXaOPjk0rRmHBaT17rAmn/BQcv24Bu3lV+JYn12AJZiAM5TvC87DaSPLrHxo+vILLwc5pM3nhb11JoRaH+smGqTgIqzadRF9XvWlRMzqggZQxDkwyW/0qm4ifrB68LtwrWdopPrNpEWdi9W2M2nP+hhUPAzTSj4E3yXErztqiQrTUhOt7ovMLE4oKO9BTQUE22BviNWm4VWYTXQG+TUSNJaw4+6OIWPlfafxlq3Gds5Dx91vRc/tyO4Rt03RQV0vz4vYaeaOYyMYBWecjSHWFWL75JntLUKmph6VTuTsPSh5EI2EYlDeKGoY/VylNWKYt07JKJAoV7t7cfp6COrYcczSw2jdO7wvYxVrhu5bSykOZknlZyunEciPDlqmDFPT2f1jodUrow1cLE+9biHF+BXT+hoWZuStoC1VR9rl3yE5sflymDUcIuKy8uPAvb8qE+sCA9rqHyqFX8qfgJGfC9/9tdTUlg+YbrTlsL8x6WtFgKw3RtL1KZ0Gatt5G8JamYkhCKruEuBavVVnjzQphIIrmIV5LD/kKLs

Also, find f(c(t)).

f(c(t)) = f(2et, $ct2, $at) =

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
Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Evaluate the line integral using u-substitution.

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
Correct.
Incorrect.