Applications of Integration

Applications of Integration

Areas Under Curves

• Integration can be used to calculate the area under a curve between the curve and the x-axis. This is known as definite integration. The area is positive if the curve is above the x-axis, and negative if it’s below.

• To do this, find the integral of the function that represents the curve, then subtract the value of the integral at the start of the interval from the value at the end.

Volumes of Revolution

• If you rotate a curve about the x-axis or y-axis, it forms a 3D shape called a solid of revolution. You can use integration to calculate the volume of this solid.

• To find the volume of a solid of revolution, you use the disc method which integrates πy^2 with respect to x or the shell method which integrates 2πx*y with respect to y.

Physical Applications

• In physics, integration is used frequently for calculations involving motion. Acceleration is the derivative of velocity and velocity is the derivative of displacement, so to find velocity given acceleration, or displacement given velocity, you have to integrate.

• It could be used to find the total distance travelled by an object, by integrating the speed function.

Economic Applications

• Economists use integration to find consumer and producer surplus, which are areas between supply and demand curves.

• It is also used in calculating the total cost given a cost function, or the total revenue given a revenue function.

Remember, practice is crucial for understanding applications of integration. Especially in the context of word problems, understanding what each integral represents can help you set up and solve the problem correctly.