Worked example: area between curves | AP Calculus AB | Khan Academy

TL;DR

Use calculus to find the area between curves by setting up a definite integral with the upper and lower bounds and evaluating it.

Transcript

• [Instructor] What we're going to do using our powers of calculus is find the area of this yellow region and if at any point you get inspired, I always encourage you to pause the video and try to work through it on your own. So, the key here is you might recognize hey, this is an area between curves. A definite integral might be useful, so I'll ju... Read More

Key Insights

• 💐 The area between curves can be calculated using definite integrals by subtracting the lower bound from the upper bound.
• 👈 The left and right boundaries of the region are determined by the points where the two curves intersect.
• 😘 The upper and lower bounds are defined by the equations of the curves themselves.
• ☺️ The definite integral allows for the calculation of the area by summing up infinitely small areas as we move along the x-axis.
• 🍹 The concept of the definite integral in finding the area between curves is based on the Riemann sum.
• ☠️ The area between curves can also represent the accumulation of quantities over a certain interval when the curves represent rates of change.
• ✊ The area between curves can be found using calculus concepts such as the reverse power rule.

Questions & Answers

Q: How do you determine the left and right boundaries of the region when finding the area between curves?

The left and right boundaries are determined by the points where the two curves intersect. These points can be found by setting the two equations equal to each other and solving for the x-values.

Q: Why subtract the lower bound from the upper bound when finding the area between curves?

Subtracting the lower bound from the upper bound gives the difference in y-values between the two curves at each point along the x-axis. This difference represents the vertical height of the region that needs to be integrated to find the area.

Q: What is the significance of setting up a definite integral to find the area between curves?

The definite integral allows us to sum up an infinite number of infinitesimally small areas to find the total area between curves. It takes into account the changing shape and position of the region as we move along the x-axis.

Q: Is it possible for the upper and lower bounds to be different curves in finding the area between curves?

Yes, the upper and lower bounds can be different curves. The area is still found by subtracting the lower bound from the upper bound to calculate the difference in y-values between the two curves at each point along the x-axis.

Summary & Key Takeaways

• The video explains how to find the area of a region between two curves using the concept of definite integrals.

• The left and right boundaries of the region are determined by the points where the two curves intersect.

• The upper and lower bounds are defined by the curves themselves, and the area is found by subtracting the lower bound from the upper bound.