Find the Area of the Region Bounded by the Graphs f(x) = 1/(6(1 + x^2)) and g(x) = (1/12)x^2

TL;DR

This video explains how to find the area bounded by two graphs using graphing and integration techniques.

Transcript

and this problem we're going to find the area bounded by these graphs let's go ahead and work through this so the hardest part will be to graph these two functions so first note that the graph of 1 over 1 plus x squared let's ignore the 6 for now see we had this the general shape of this graph is something like this you might say how do you know th... Read More

Key Insights

• 😥 The general shape of the graph of a rational function, such as 1/(1+x^2), can help in estimating the points of intersection with other functions.
• 😥 Factoring a quartet equation can be used to find the points of intersection between two functions.
• 📈 Integration can be used to find the area bounded by two graphs when the functions intersect.

Questions & Answers

Q: How do you find the points of intersection between two functions?

To find the points of intersection, set the two functions equal to each other and solve for x. In this case, we set 1/(1+x^2) = 1/12x^2 and solve the resulting quartet equation to find the points of intersection as 1 and -1.

Q: Why is the graph of 1/(1+x^2) shaped like a curve?

The graph of 1/(1+x^2) is a curve because it is a specific type of function called a rational function. This type of function often has curves on its graph, and through experience and observation, we know the general shape of this curve.

Q: How do you find the area bounded by two graphs?

To find the area bounded by two graphs, you can use integration. By setting up an integral with the top graph minus the bottom graph and integrating over the interval where the two graphs intersect, you can find the desired area.

Q: Why did the video suggest integrating from 0 to 1 instead of from -1 to 1?

The video suggested integrating from 0 to 1 because of the symmetry of the graphs. The area between -1 and 0 would be the same as the area between 0 and 1, so by integrating only from 0 to 1 and then multiplying the result by 2, we obtain the total area bounded by the graphs.

Summary & Key Takeaways

• The video demonstrates how to graph two functions, 1/(1+x^2) and 1/12x^2, and find their points of intersection.

• By setting the two functions equal to each other, factoring a quartet equation, and solving for x, the points of intersection are found to be 1 and -1.

• The video then explains how to use vertical rectangles and integration to find the area bounded by the two graphs, resulting in a final answer of 0.206.