Introduction to the line integral | Multivariable Calculus | Khan Academy

TL;DR

Learn how to calculate the area under a curve in three dimensions using the definite integral.

Transcript

If we're just dealing with two dimensions, and we want to find the area under a curve, we have good tools in our toolkit already to do it, and I'll just remind us of our tools. so let's say, that's the x-axis, that's the y-axis, let me draw some arbitrary function right here, and that's my function f of x. And let's say we want to find the area bet... Read More

Key Insights

• 👈 The area under a curve in two dimensions is calculated by multiplying small changes in x by the value of the function at that point and summing them all up using a definite integral.
• ✖️ In three dimensions, the area under a curved path can be calculated by parameterizing the x and y variables and multiplying small changes in arc length by the value of the function at that point and summing them all up using a definite integral.

Questions & Answers

Q: How can the area under a curve in two dimensions be calculated?

In two dimensions, the area under a curve can be calculated by multiplying small changes in x (dx) by the value of the function (f(x)) at that point and summing them all up using a definite integral.

Q: How is the area under a curve in three dimensions different from two dimensions?

In three dimensions, the area under a curved path can be calculated by parameterizing the x and y variables, multiplying small changes in arc length (ds) by the value of the function (f(x,y)) at that point, and summing them all up using a definite integral.

Q: How can the formula for the area under a curve in three dimensions be simplified?

By manipulating differentials, the formula for the area under a curve in three dimensions can be simplified to include only derivatives of x and y with respect to t (the parameter), which can be evaluated using a definite integral.

Q: How is the formula for the area under a curve in three dimensions expressed?

The formula for the area under a curve in three dimensions is expressed as the definite integral of f(x(t), y(t)) times the square root of (dx(t)/dt)^2 + (dy(t)/dt)^2 with respect to t, where x(t) and y(t) are functions of t.

Summary & Key Takeaways

• In two dimensions, the area under a curve can be found by multiplying small changes in x (dx) by the value of the function (f(x)) at that point and summing them all up.

• In three dimensions, the area under a curved path can be found by parameterizing the x and y variables and multiplying small changes in arc length (ds) by the value of the function (f(x,y)) at that point and summing them all up.

• By manipulating differentials, the formula for the area under a curve in three dimensions can be simplified to include only derivatives of x and y with respect to t (the parameter) and can be evaluated using a definite integral.