maths calculation help pleas!!?

Question

Find the area bound by the follwoing curves, the x-axis and the indicated lines

1) y= x between x=1 and x=4
2) y = 2x / x2 - 3 between x=2 and x=3
3) y = e^x + 2 between x = 0 and x = 2
4) y = 2x^3 - 4x + 7 between x=1 and x=3

please can you show me the steps to work this out

thank you

Answer 1

To find the area under a curve you need to work out the integral of the function describing the curve between the limits given for x.



So for 1) you need ∫xdx between limits 1 and 4



for 2) you need ∫2x/(x^2 - 3)dx between limits 2 and 3



for 3) you need ∫(e^x + 2)dx between limits x =0 and x = 2 and so on.



No need to draw a graph and shade areas!!



I'll do Qn 4 for you;



Area = ∫(2x^3 - 4x + 7)dx



Area = (2x^4/4 - 4x^2/2 + 7x) between x = 1 and x =3 [note no constant of integration C, since you are finding a definite integral]



Area = (x^4/2 - 2x^2 + 7x) between x = 1 and x =3



Area = (3^4/2 - 2*3^2 + 7*3) - (1^4/2 -2*1^2 + 7*1)



Area = (81/2 - 18 + 21) - (1/2 - 2 +7)



Area = 40 + 3 - 5 = 38





Hope this helps.

Answer 2

Theory : - Given y = f(x) ...and to find area covered by the curve y with x axis between point x=a and x=b is ...just the value of integration :-



Limit of x from a---> b [ y.dx ]



---



1) limit of x from 1 to 4 [ x.dx ] = limit of x from 1 to 4 [ x^2/2 ] = 16/2 - 1/2 = 15/2 ans .



2) limit of x from 2 to 3 [ 2x / (x^2 - 3 ) dx ] = limit of x from 2 to 3 [ ln(x^2-3) ] = ln ( 6 ) ans .



Note :- Integration of 2x/(x^2-3) is calculate by substituting x^2-3 as t

So it becomes 2x.dx / (t) .. and t = x^2-3 So dt/dx = 2x Hence 2x.dx = dt



Therefore ... dt/t =====> ln(t) ..and hence ln(x^2-3)



3) limit of x from 0 to 2 [ (e^x + 2).dx ] = limit of x from 0 to 2 [ e^x + 2x ] = [ e^2 + 4 - 1 ] = [e^2 +3]



Note :- Integration of e^x = e^x



4) limit of x from 1 to 3 [ 2x^3 -4x + 7 ] = limit of x from 1 to 3 [ x^4/2 - 2x^2 + 7x ]

============================= after putting limit [ 3^4/2 - 2*3^2 + 7*3 - ( 1/2 - 2*1 + 7 ) ]

=============================ans is 87/2 - 11/2 = 38





I hope it is clear now....if not then ask further



Regard

Yagya

Answer 3

You basically need to draw a graph plotting each of these lines on it the area found inside of the limits set by these lines is what the question is asking for. e.g. for 1) where y=x u need to draw a line straight 0 at a 45 degree angle as this is the line for y=x but only draw this line for values of x=1 to x=4 do the same with the other lines and you will be left with a shape inside these lines that you should shade to show that it is the area in question. Online graph calculators are good to show you how to represent functions on graphs.

Hope this helps!

Ben