Calculus Online: Lab 3 Solutions


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Question 1: Arc Length

2 marks: We asked you to estimate the length of the cable by approximating the cable by straight lines. You should have found an answer near 6.19 or 6.20.


Question 2: Numerical Techniques

Part a (2 marks)

You were asked to position the dots so that \[ L_2 = h(y_0 +y_1) = 5. \] There are many ways to accomplish this task.

Part b (2 marks)

You were asked to position the dots so that

\[ T_2 = \frac h2(y_0 + 2y_1 + y_2) = 5. \]
Once again, this can be done in many ways.

Part c (2 marks)

You were asked to position the dots so that

\[ S_2 = \frac h3(y_0 + 4y_1 + y_2) = 3. \]


Question 3: Pi

In this question, use our three approximations on the definite integral

\[ \int_0^1 \frac{4}{1+x^2}~dx = 4\tan^{-1}x ~|_0^1 = 4\tan^{-1}(1) = \pi. \]

Part a (2 marks)

The left endpoint rule gives an approximation of 3.6.

Part b (2 marks)

The trapezoidal rule gives an approximation of 3.1.

Part c (2 marks)

Simpson's rule gives an approximation of 3.133.