Problem 12 - 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. $$ \int_{-2}^{3}\left[(y+6)-y^{2}\right] d y $$

Step-by-Step Solution

Answer

The shaded region is the area bounded by the curves of the functions \( y+6 \) and \( y^{2} \) from x=-2 to x=3. The area under the line and above the parabola for these x-values is represented by the given definite integral.

Step 1: Sketch Graphs

Sketch the graph for each function, \( y+6 \) (a straight line) and \( y^{2} \) (a parable). Use an appropriate scale for both x and y to clearly represent the key points of each function.

Step 2: Identify Bounded Area

Identify the area bounded by the two curves. Remember that the limits of the the definite integral (-2 to 3) indicate the x-values for which the area is defined. In this case, the area under the line \( y+6 \) and above the parabola \( y^{2} \) from x=-2 to x=3 is being sought.

Step 3: Shade the Region

Shade the region that is bounded by these two functions from x=-2 to x=3. This shaded area is the one represented by the given definite integral.