find the total area under the curve

Question

Find the area under the curve from  to  and evaluate it for , .

Then find the total area under this curve for .

I know that (t=10) = 9/1600 and that (t=100) = 621/40000,,,, but I don't know the total area. Please help.

Answer 1

the area under y = 2x-3 from x= 8 to x = t

is the area of a quadralateral with vertices (8,0), (t,0), (8,13) and (t, 2t-3)

which equals the area of a triangle on top of the area of a rectangle below the triangle and above the x axis

It may help to sketch a rough graph, using t as a number larger than 8, such as 10

then the area under the curve (which is a straight line with slope = 2) is bh/2 for the triangle plus 2 times 13

where b=2 and h= 17-13 =

Total area = 2(8)/2 + 2(13) = 4+26 = 30

but in general for any x=t, the total area has base b = t-8 and the height of the triangle would be h=2t--16

with height of the rectangle as 13. Total area = (t-8)^2 + (t-8)(13)=

= t^2 -3t -40

for t= 100

the area = 100^2 -300- 40 = 10,000-340 = 9,660

for t = 10

the area= 10^2 -3(10) -40 = 100-30-40= 30

That's a geometric method, giving the area

for calculus take the integral of 2x-3 = x^2 -3x

evaluate the integral between 8 to t

t^2 -3t -64+24

= t^2 -3t - 40

for t= 10

the area = 100-30-40 = 30

for t=100

the area = 10000-300-40 = 9,660

if you may have meant evaluate from t=10 to t=100

then that area under the line is 9660-30 = 9630