Integrating scaled version of function | AP Calculus AB | Khan Academy

TL;DR

The area under a scaled version of a function is equal to the original function's area multiplied by the scaling factor.

Transcript

• [Voiceover] We've already seen and you're probably getting tired of me pointing it out repeatedly, that this yellow area right over here, this area under the curve y is equal to f of x and above the positive x-axis or I guess I can say just above the x-axis between x equals a and x equals b, that we can denote this area right over here as the def... Read More

Key Insights

• 🧑‍🏭 The area under a scaled version of a function is equal to the scaling factor multiplied by the area under the original function.
• 🧑‍🏭 Scaling a shape by a factor c scales up its area by the same factor c.
• 📤 The definite integral from a to b of cf(x) dx represents the scaled area under the curve y=cf(x) between x=a and x=b.

Questions & Answers

Q: What does the yellow area represent in the video?

The yellow area represents the definite integral from a to b of f(x) dx, which is the area under the curve y=f(x) between x=a and x=b.

Q: How does the scaled version of f(x) differ from the original function?

The scaled version of f(x), y=c*f(x), is a vertical scaling of the original function. The height of every point on the curve is multiplied by the scaling factor, c.

Q: What is the relationship between the green area and the yellow area?

The green area, under the scaled curve y=cf(x), is equal to the definite integral from a to b of cf(x) dx. It is a scaled-up version of the yellow area, with the scaling factor, c, multiplying the original area.

Q: How does scaling affect the area under a curve?

Scaling the vertical dimension of a shape, in this case, the function f(x), by a factor c, scales up the area under the curve by the same factor c. This means that the area under the scaled curve y=c*f(x) is c times the area under the original curve y=f(x).

Summary & Key Takeaways

• The yellow area under the curve y=f(x) between x=a and x=b is denoted as the definite integral from a to b of f(x) dx.

• Exploring the area under a scaled version of f(x), y=cf(x), reveals that the green area is equal to the definite integral from a to b of cf(x) dx.

• The area under the scaled curve is a scaled-up version of the area under the original curve, with the scaling factor, c, multiplying the original area.