Properties of Definite Integrals

The definite integral is the limit and summation that we use to obtain the net area between a function and the x-axis in the previous section. It’s also worth noting that the definite integral’s notation is remarkably similar to that of an indefinite integral. Soon you will understand the reason for this.

The lower limit of the integral is the number “a” at the bottom of the integral sign, and the upper limit of the integral is the number “b” at the top of the integral sign. Furthermore, even though a and b were presented as an interval, the lower limit does not have to be smaller than the upper limit. The integration interval is sometimes referred to as [a,b](taken together).

Properties of definite integrals

Following is the list of some properties of definite integrals which are easy to read and understand.

Property 1: p∫q f(a) da = p∫q f(t) dt

Property 2: p∫q f(a) da = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0

Property 3: p∫q f(a) da = p∫r f(a) d(a) + r∫q f(a) d(a)

Property 4: p∫q f(a) da = p∫q f( p + q – a) d(a)

Property 5: 0∫p f(a) d(a) = 0∫p f(p – a) d(a)

Property 6: ∫02p f(a)da = ∫0p f(a)da +∫0p f(2p-a)da…if f(2p-a) = f(a)

Property 7: 2 parts

1. ∫02p f(a)da = 2∫0p f(a) da … if f(2p-a) = f(a)

2. ∫02p f(a)da = 0 … if f(2p-a) = -f(a)

Property 8: 2 parts

1. ∫-pp f(a)da = 2∫0p f(a) da … if f(-a) = f(a) or it’s an even function

2. ∫-pp f(a)da = 0 … if f(-a) = -f(a) or it’s an odd function

Evaluating Definite Integrals

As the name implies, integration brings two or more things together. Integration is the procedure of integrating functions in mathematics. The alternate word for integration is a summation, which adds up the complete function or graphically finds the area under the curve function. Evaluating integrals is the inverse of differentiation, breaking down a function into smaller functions. In contrast, integration adds smaller portions to get the total area under the curve. 

Types of definite integrals 

1. Definite integrals involving rational or irrational expressions

2. Definite integrals involving trigonometric functions

3. Definite integrals involving exponential functions

4. Definite integrals involving logarithmic functions

5. Definite integrals involving hyperbolic functions

Conclusion

The definite integral combines all of the Riemann sums we learned about in the last session. This lesson aims to present and debate the definition of the definite integral and familiarise yourself with its basic properties. The general area of geometric regions will be used to evaluate definite integrals in this lesson’s talks.