Problem 14 - Essential Calculus

Question

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=-\frac{3}{8} x(x-8), y=10-\frac{1}{2} x, x=2, x=8 $$

Step-by-Step Solution

Answer

The area of the region bounded by the curves is 18 square units.

Step 1: Sketch the Graphs

First, sketch the region described in the problem. Plot the quadratic function \(y=-\frac{3}{8} x(x-8)\), the linear function \(y=10-\frac{1}{2} x\), and the vertical lines \(x=2\), \(x=8\). This graph will provide a visualization of the region whose area is to be calculated.

Step 2: Find Intersection Points

Find the points where the graphs intersect. These points can be computed by setting the equations equal to each other and solving for \(x\) and \(y\) respectively. For the curves to intersect, both \(x\) and \(y\) values would be the same. So, -\frac{3}{8} x(x-8) = 10-\frac{1}{2} x and solve for \(x\) which will give \(x=5\) and \(x=2\). Substituting these into the equation of the line will yield respective \(y\)-values.

Step 3: Calculate the Area

To calculate the area of the region, take the integral of the absolute difference of the two functions between the points of intersection. Specifically, the area \(A\) equals to \(\int_{2}^{8} (10-\frac{1}{2}x-[-\frac{3}{8}x(x-8)])dx\). Applying the rules of integration, calculate the definite integral to get the area.