Problem 3 - Calculus Early Transcendentals

Question

Make a sketch to show a case in which the area bounded by two curves is most easily found by integrating with respect to \(x\).

Step-by-Step Solution

Answer

Question: Find the area bounded by the curves \(y = x^2\) and \(y = x\) by integrating with respect to x. Answer: The area bounded by these two curves is \(\frac{1}{6}\).

Step 1: Select equations of two curves

For this example, let's choose the functions \(y=x^2\) and \(y=x\).

Step 2: Find the intersection points

In order to find the area between the two curves, we first need to find the points where the curves intersect. Set the two function equal to each other to find the intersection points: \(x^2 = x\) \(x^2 - x = 0\) \(x(x-1) = 0\) This gives us two intersection points, x = 0 and x = 1. The corresponding y values for these points can be found by plugging the x values back into either of the equations (let's use \(y = x^2\)): For x = 0: \(y = (0)^2 = 0\) For x = 1: \(y = (1)^2 = 1\) The intersection points are (0, 0) and (1, 1).

Step 3: Sketch the curves

Draw the two curves on the Cartesian plane, representing the functions \(y = x^2\) and \(y = x\), and plot the intersection points found in Step 2. The first curve is a parabola opening upwards with its vertex at the origin (0, 0), while the second curve is a diagonal straight line passing through the origin with a positive slope of 1. When viewing the graph, it is apparent that the area we want to calculate is between these two curves from x = 0 to x = 1.

Step 4: Set up the integral for the area

Since we are integrating with respect to x, we set up the integral using the difference between the two functions over the given interval: \(\int_{0}^{1} (x - x^2) dx\)

Step 5: Calculate the area

Evaluate the integral to find the area between the curves: Area = \(\int_{0}^{1} (x - x^2) dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1\) Area = \(\left[\frac{(1)^2}{2} - \frac{(1)^3}{3}\right] - \left[\frac{(0)^2}{2} - \frac{(0)^3}{3}\right]\) Area = \(\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\) To summarize, we have chosen two curves \(y = x^2\) and \(y = x\), found their intersection points (0, 0) and (1, 1), sketched the curves on the Cartesian plane to visualize the area between them, and set up and evaluated the integral for the area with respect to x. The final solution is that the area bounded by these two curves is \(\frac{1}{6}\).