Problem 40 - Calculus Early Transcendentals

Question

Evaluate the following integrals or state that they diverge. $$\int_{3}^{4} \frac{d z}{(z-3)^{3 / 2}}$$

Step-by-Step Solution

Answer

If it converges, what is its value? Answer: The integral diverges.

Step 1: Identify the function and integration limits

We are given a definite integral with the function \(f(z) = \frac{1}{(z-3)^{3/2}}\) and integration limits \(3\) and \(4\). The integral is as follows: $$\int_{3}^{4} \frac{d z}{(z-3)^{3/2}}$$

Step 2: Rewrite the function as a power function

To perform integration more easily, we will rewrite the given function in the form \(f(z)= (z-3)^n\). In this case, we have \(n = -\frac{3}{2}\). Therefore, our integral becomes: $$\int_{3}^{4} (z-3)^{-3/2} dz$$

Step 3: Integrate the power function

Now, we can use the power rule for integration, which states that \(\int z^n dz = \frac{z^{n+1}}{n+1}\), where \(n \neq -1\). Since \(n=-\frac{3}{2}\), we can apply this rule and integrate our function. $$\int (z-3)^{-3/2} dz=\frac{(z-3)^{-3/2 +1 }}{-3/2 +1}$$ Simplify the expression: $$=\frac{2}{1}(z-3)^{-1/2}$$

Step 4: Evaluate the definite integral

Finally, we need to evaluate the definite integral using the integration limits \(3\) and \(4\). We will do this by calculating the antiderivative at the upper limit, minus the antiderivative at the lower limit: $$\Big[2(z-3)^{-1/2}\Big]_{3}^{4}$$ Plug in the values and subtract: $$=2(4-3)^{-1/2}-2(3-3)^{-1/2}$$ This expression simplifies to: $$=2(1)^{-1/2}-2(0)^{-1/2}$$ Since division by zero is undefined, we conclude that the integral diverges.