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JA8. Exercises on Integration

Statement

Write 2 true/false questions that illustrate a variety of ideas from this week’s topics that you might put on this exam if you were teaching the class.

  • Give a key
  • Explain the answers
  • Explain why you chose these particular questions and what the questions will assess

Solution

Example 1

Statement: If the integral of a function f(x) from a to b is zero, it implies that f(x) is zero everywhere in [a, b].

According to the net change theorem (Herman & Strang, 2020, p. 567), the integral of a function f(x) from a to be can be found according to the following formula:

\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

The right side of equation is zero when F(b) - F(a) = 0, which means that the integral which may mean one of the following:

  • The function f(x) did not change from a to b, which means that f(x) is zero everywhere in [a, b].
  • The function f(x) changed from a to b, from one side to another to the point that the change of one side cancels out the change of the other side.

Thus the statement is false as there are more possible scenarios than the one described in the statement.

This was a good question to think about the meaning of integral is zero; which means either the function did not move at all, or moved and then went back to the original position.

Also, it is good to think that these kind of questions depend on the context of the problem, and it what may be true in one context may not be true in another.

Example 2

Statement: The integral of a constant function c over the interval [a, b] is equal to c times the length of the interval.

According the above equation, the integral of a constant function c over the interval [a, b] is:

As we know that the antiderivative of a constant function f(x) = c is F(x) = cx, we can rewrite the equation as:

\[ \begin{aligned} \int_{a}^{b} c \, dx &= F(b) - F(a) \\ &= c(b) - c(a) \\ &= cb - ca \\ &= c(b - a) \end{aligned} \]

We can see that b-a represents the length of the interval, and thus the statement is true.

This is also a good question as it gives the impression that the rule is also applied to the constant functions as well.

Conclusion

The text had given two examples of two questions that we might add to the exam, with the relevant explanations. One statement is true and the other is false. I took me a fair amount of time to generate the two questions which is a chance to thank you for your effort in preparing the exams; and I wish the questions in the real exam are easier than these two.

References