April 14, 2024

Definite integral

If there are limit values indicated on your integral, you are looking to find a definite integral. This means that these values will be used to find a numeric answer rather than a function.

This is done in the following way, where the values for a and b are substituted as x-values into your indefinite integral:

Definite integral
Be careful, the order you substitute a and b into the indefinite integral is relevant for your answer:
Definite integral

Solving definite integrals

Solving definite integrals
  1. Find the indefinite integral (without +C)
Solving definite integrals
2. Fill in: F(b) − F(a) (integral x = b) − (integral x = a)
Solving definite integrals

5.2.1  Area

Area between a curve and the x-axis

Area between a curve and the x-axis
DB 5.5
By determining a definite integral for a function, you can find the area beneath the curve that is between the two x-values indicated as its limits.
Area between a curve and the x-axis
Area between a curve and the x-axis
The area below the x-axis gives a negative value for its area. You must take that value as a positive value to determine the area between a curve and the x-axis. Sketching the graph will show what part of the function lies below the x-axis. So
Area between a curve and the x-axis
Area between two curves
Area between two curves
Using definite integrals you can also find the areas enclosed between curves:
Area between two curves
With g(x) as the “top” function (furthest from the x-axis). For the area between curves, it does not matter what is above/below the x-axis.

Finding areas with definite integrals.

Let y = x3 − 4x3 + 3x Find the area from x = 0 to x = 3.
  1. Find the x-intercepts: f (x) = 0

x3 − 4x3 + 3x = 0, using the GDC: x = 0 or x = 1 or x = 3

2. If any of the x-intercepts lie within the range, sketch the function to see which parts lie above and below the x-axis.

areas with definite integrals

3. Setup integrals and integrate

areas with definite integrals
4. Add up the areas (and remember areas are never negative!)
areas with definite integrals

Alternatively, use the calculator to find areas

Alternatively, use the calculator to find areas
Alternatively, use the calculator to find areas
In this case, the area is 3.083