Question

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{1}^{5} x^{2} \sqrt{x-1} d x $$

Short answer

The answer is the calculated definite integral, and it represents the area under the curve \( f(x) = x^{2}\sqrt{x-1} \) from \( x = 1 \) to \( x = 5 \).

Step-by-step solution

Plotting the function

First, the function \( f(x) = x^{2}\sqrt{x-1} \) should be plotted using a graphing utility from x = 1 to x = 5 to understand the shape of the function and the area which we are trying to find.

Calculating the definite integral

Secondly, using the graphing utility or calculator compute the area under the curve by calculating the definite integral \( \int_{1}^{5} x^{2}\sqrt{x-1} \, dx \). Some graphing utilities provide the functionality to calculate the definite integral directly.

Interpret the results

Finally, make an interpretation of the results. The answer obtained from Step 2 gives the area under the curve of the function from x=1 to x=5. The visual interpretation is the shaded region under the curve of the function on the plotted graph from x=1 to x=5.