## Question

###### How do you evaluate the definite integral by the limit definition given #int xdx# from [-2,3]?

## Answers

Definite integral

#int_(-2)^3xdx=5/2# ## Explanation:

A definite integral is an integral with upper and lower limits. The definite integral of

#f(x)# is a number and it represents the area under the curve#f(x)# from#x=a# to#x=b# .Its value is equal to

#F(b)-F(a)# , where#F(x)=intf(x)dx# and this is expressed as#int_a^bf(x)dx=[F(x)]_a^b# As

#intxdx=x^2/2+c#

#int_(-2)^3xdx=[x^2/2+c]_(-2)^3# =

#[(3)^2/2+c-((-2)^2/2+c)]# =

#[9/2+c-4/2-c]# =

#5/2# Observe that constant term

#c# has cancelled out. As such we can avoid adding#c# to#F(x)# , while calculating definite integral.Further see the following figure depicting area under the curve

#f(x)=x# considered above.Observe that area under the

#x# -axis (shown as pink in colour) has been taken as negative. If in some application of this concept, this is to be taken as positive, one must take adequate care.