Problem 6 - Calculus Early Transcendentals

Question

If the change of variables \(u=x^{2}-4\) is used to evaluate the definite integral \(\int_{2}^{4} f(x) d x,\) what are the new limits of integration?

Step-by-Step Solution

Answer

Question: Determine the new limits of integration after the substitution \(u = x^2 - 4\) given that the original limits of integration are \(x=2\) (lower limit) and \(x=4\) (upper limit). Answer: The new limits of integration in terms of the variable \(u\) are \(u = 0\) (new lower limit) and \(u = 12\) (new upper limit).

Step 1: Substitute the limits of integration for x into the transformation

Let's begin with finding the equivalent limits of integration for \(u\). To do that, we'll substitute the given limits of integration for \(x\) into the transformation equation \(u = x^2 - 4\). For the lower limit \(x=2\), we have: \(u = (2)^2 - 4\) For the upper limit \(x=4\), we have: \(u = (4)^2 - 4\)

Step 2: Evaluate u for both limits of integration

Now, we will simplify and evaluate these expressions for both limits of integration. For the lower limit, we get: \(u = 2^2 - 4 = 4 - 4 = 0\) For the upper limit, we have: \(u = 4^2 - 4 = 16 - 4 = 12\)

Step 3: Report the new limits of integration

Now that we have found the corresponding limits of integration for \(u\), we can write them: The new limits of integration will be \(u = 0\) (new lower limit) and \(u = 12\) (new upper limit), so the definite integral with the change of variable becomes: \(\int_0^{12} f(u) du\)