Finding The Area Bounded By Three Curves Using Definite Integrals - Calculus

TL;DR

Learn how to set up integrals to find the area bounded by three curves.

Transcript

in this video we're going to talk about how to set up an integral that will help us to find the area that's bounded by three curves or three graphs so we're given the equation of the three graphs but we don't know the points of intersection so the first thing we need to do is we need to draw a rough sketch of these graphs so y equals x^2 we know it... Read More

Key Insights

• 🤪 Three curves are given: y = x^2 (opens wider), y = 1/4x^2 + 2 (shifted up), and y = -1/2x + 2 (line going down).
• 😥 Points of intersection are found by setting the equations of the curves equal to each other.
• 🗂️ The area between the three curves is divided into two separate regions.

Questions & Answers

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

To find the points of intersection, set the two equations equal to each other and solve for x. This will give you the x-values where the curves intersect.

Q: What is the formula to find the area between two curves?

The formula to find the area between two curves is the definite integral of the top function minus the bottom function, integrated over the interval of intersection.

Q: Can you estimate the area even if the graph is not drawn to scale?

Yes, as long as you can identify the points of intersection and the shaded region, you can still calculate the correct area using the integral expressions.

Q: What does it mean when a curve opens wider or narrower?

When a curve has a coefficient less than one in front of the variable, it opens wider. When the coefficient is greater than one, it opens narrower.

Summary & Key Takeaways

• The video explains how to sketch and identify three curves given by equations.

• The points of intersection of the curves are found using algebraic methods.

• The area bounded by the curves is calculated using definite integrals.