evaluate definite integrals by using areas: int[0,b](x/2)dx

[annajee]

Use areas to evaluate the integrals:

#1: integral{0 to b} (x / 2) dx, b > 0

#2: integral{a to b} 2s ds, 0 < a < b

I do not know what it means by "use areas". Can you tell me what it means to use areas to do this and how do you do that.

Thank you

 

[Deleted member 4993]

Re: evaluating definite integrals

go to google.com

type in:

definite integral by area

You'll find ~295000 sites ready to help you with example problem.

Please show us your work, indicating exactly where you are stuck, so that we know where to begin to help you.

 

[fasteddie65]

Re: evaluating definite integrals

#1: integral{0 to b} (x / 2) dx, b > 0

Graphing this equation, you will see that it is right triangle, with base b and height b/2.
A = (1/2)bh = (1/2)b(b/2) = b^2/4

Using a definite integral would produce the same answer.


#2: integral{a to b} 2s ds, 0 < a < b

Graphing this equation, you will see that it is a trapezoid, with lower base 2a and upper base 2b, and height (b - a).
A = (1/2)h(b1 + b2) = (1/2)(b - a)(2a + 2b) = (1/2)(2)(b - a)(b + a) = b^2 - a^2