Find the area bounded by the x-axis, the curve y=e^2x and the ordinates X=4 and X=6.?

Answer 1

Finally a calculus question time to have some fun my friend

so the way to solve this problem is integrate

integral((e^2x)dx) with upper limit = 6 and lower limit = 4
to do this make a substution.
let u = 2x then du = 2dx or du/2 = dx and you also havce to change your upper and lower limit. So your Upper limit equals u(6) = 2(6) = 12 and lower limit equals u(4) = 2(4) = 8
So now your new integral is
integral((e^u)du/2) upper limit = 12 and lower limit = 8 now you can pull out the 1/2 since it is a constant which gives
1/2 integral((e^u)du) now when you integrate a e^x function you get the same thing so in our case we have
1/2[e^u] upper limit 12 and lower limit 8 which gives
1/2[e^12 - e^8] and that is your answer

Answer 2

Hi,

I typed y1 = e^(2x) into my TI-83calculator. Then I went to the CALC commands on the top row and chose #7 integral f(x) dx. It asked for a lower bound and I typed in 4. Then it asked for an upper bound and I typed in 6. Once that was entered the calculator shaded that area on the screen and told me the integral worked out to 79,886.917.

I hope that helps.

Answer 3

so the way to solve this problem is integration

integral((e^2x)dx) with upper limit = 6 and lower limit = 4
to do this make a substution.
let u = 2x then du = 2dx or du/2 = dx and you also havce to change your upper and lower limit. So your Upper limit equals u(6) = 2(6) = 12 and lower limit equals u(4) = 2(4) = 8
So now your new integral is
integral((e^u)du/2) upper limit = 12 and lower limit = 8 now you can pull out the 1/2 since it is a constant which gives
1/2 integral((e^u)du) now when you integrate a e^x function you get the same thing so in our case we have
1/2[e^u] upper limit 12 and lower limit 8 which gives
1/2[e^12 - e^8] and that is your answer

On the calculator the integral worked out to be 79,886.917

Answer 4

y = e^(2x), x = 4, x = 6

We know for a fact that y = e^(2x) doesn't cross the x-axis, so we don't have to worry about creating two integrals.

The bounded area means our integral bounds are x = 4 to 6, and our area is

A = Integral (4 to 6, e^(2x) dx )

Integrating, we get

A = (1/2)e^(2x) {evaluated from 4 to 6}

A = [(1/2)e^(2*6) - (1/2)e^(2*4) ]

A = [(1/2)e^12 - (1/2)e^8]
A = (1/2)e^8 [e^4 - 1]

Answer 5

Hi,

I typed y1 = e^(2x) into my TI-83calculator. Then I went to the CALC commands on the top row and chose #7 integral f(x) dx. It asked for a lower bound and I typed in 4. Then it asked for an upper bound and I typed in 6. Once that was entered the calculator shaded that area on the screen and told me the integral worked out to 79,886.917.

I hope that helps.

Answer 6

A = ∫ e^(2x) dx between lims 4 and 6
A = (1/2).[ e^(2x) ] lims 4 to 6
A = (1/2).[e^12 - e^(8)]
A = (1/2).[e^8(e^4 - 1)]
A = (1/2) x 2978 x 53.6
A = 79810 units² (rounded)

Answer 7

Looks like you got some good answers. The calculations are all correct. I like the calculator answer also.

Answer 8

It will be in your maths book.Open and see that.

Answer 9

good question indeed