Problem 10 - College Algebra and Calculus: An Applied Approach

Question

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-4}^{0}\left[(x-6)-\left(x^{2}+5 x-6\right)\right] d x $$

Step-by-Step Solution

Answer

The graphs of \(y=x-6\) and \(y=x^{2}+5x-6\) intersect at two points within the interval of [-4,0]. Shade the region bounded by these functions and the vertical lines x=-4 and x=0. This represents the integral \(\int_{-4}^{0}\left[(x-6)-\left(x^{2}+5 x-6\right)\right] dx\).

Step 1: Graph the Function \(y=x-6\)

The function \(y=x-6\) is a linear function with slope 1 and y-intercept at -6. Therefore, start at the point (0,-6) on the y-axis, and draw a line that rises to the right.

Step 2: Graph the Function \(y=x^{2}+5x-6\)

This function is a quadratic function. The graph of \(y=x^{2}+5x-6\) is a parabola with its vertex given by \(-\frac{b}{2a},f(-\frac{b}{2a})\), where here a=1, b=5, and c=-6. Therefore, the vertex is at \(-\frac{5}{2}\), plug this into the function to calculate the y-coordinate. Once found, plot this point and sketch the graph of the parabola.

Step 3: Find the Intersection Points

To shade the required region, first find the intersection points of the two functions in the interval [-4,0]. Solve the equation \(x-6 = x^{2}+5x-6\) to find these points.

Step 4: Shade the Required Region

Shade the region between the two curves in the interval from -4 to 0. This is the area that the definite integral represents.