Question

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=\sqrt{1+x^{3}}, \quad y=\frac{1}{2} x+2, \quad x=0 $$

Short answer

To find the area of the region bounded by these equations, plot the graphs and determine their points of intersection. Set up and solve a definite integral over this interval where the integrand is the difference of the corresponding \(y\) values of the larger function and the smaller function. Verify the integral through a graphing utility.

Step-by-step solution

Graph the Equations

Using a graphing utility, graph the equations \(y = \sqrt{1+x^{3}}\), \(y = \frac{1}{2} x + 2\), and \(x = 0\). Observe the region bounded by these graphs.

Find Points of Intersection

Set \(\sqrt{1+x^{3}} = \frac{1}{2} x + 2\) and solve for \(x\). These will be the points of intersection.

Set Up Definite Integrals

Identify the finite boundaries of the region based on the points of intersection and \(x=0\). You will set up your definite integral to determine the bounded area as the difference of the definite integrals of \(y = \frac{1}{2} x + 2\) and \(y = \sqrt{1+x^{3}}\).

Solve the Definite Integrals

Calculate the definite integrals to get the area between the two curves from the limits found in step 2.

Verify Your Results

Use the integration capabilities of the graphing utility to verify the result from step 4. The resulting areas obtained from both methods should match.