Question

Find the area of the region between the graph of f and the x-axis on the given interval. Give both exact (symbolic) and approximate (numeric) answers.
f(x)=e^(-x)+1

Answer 1

from 0 to 8

Answer 2

first I found the antiderivative which i got to be (-1/x)(e^-x)+1x

Answer 3

then i plugged in 8 and subtracted it from 0
since when plugging in 0 everything is 0 I just plugged in 8 and got
(-1/8)(e^-8)+8
but wrong:(

Answer 4

Looks like you made a bad assumption: when plugging in 0, you can't just simply say that everything is 0. The anti-derivative doesn't exist at x=0, so you can't directly apply the FTC. Try using a different technique.

Answer 5

im not sure what technique i should use?

Answer 6

Well, first of all, are you sure you're not supposed to use a calculator on this question? It's by no means a trivial integral for a first-year calculus class.

Answer 7

ya i can use one on the homework...on my exams i cant so i try not to...but i can for here, but im not even sure how to use a calculator to solve this except for plugging in numbers?

Answer 8

If you're on a TI calculator, you use the fnInt function to take definite integrals. For this function, the syntax would be fnInt(e^(-X)+1,X,0,8) on an older calculator, or if you have a newer one, an integral symbol will pop up and you can put in the numbers/function where they all belong.

Answer 9

This isn't impossible without a calculator, of course; I'm just guessing that it's intended that you use one for this problem.

Answer 10

where is the fnlnt button? sorry i dont see it

Answer 11

Go to the Math menu, you'll find it towards the bottom there.

Answer 12

is the reason it can only be done on a calculator bc the 0? bc we have done indefinite integral problems plenty from 0 to a number, and thats what I always do and most of the time im right

Answer 13

It's not because of the zero; it's because of the division by zero. There are some instances in which integrals actually don't even exist, or "don't converge," and so sometimes divisions by zero come up there - but be careful, because division by zero doesn't imply nonexistence. In this case, the answer will be very close to 9 (which is very far from infinity). Again, it can be done without a calculator, but not without some fancy integration techniques.

Answer 14

i got the right answer but i need to plug it in the exact number as well (like with exponents and e and ln or whatever other number/symbols are in it) is there a way i can convert my numerical answer to that on the calculator?

Answer 15

if \[\large f(x)=e^{-x}+1\]the antiderivative is

Answer 16

(-1/x)e^-x+x?

Answer 17

\[\large F(x)=-e^{-x} + x + C\] you can drop the constant C in a definite integral problem.

Answer 18

The area in exact form is F(8) - F(0)

Answer 19

so it should be
((-1/8)e^(-8)+8)-((-1/0)?

Answer 20

bc e^0 is 1 and 0 is 0
?

Answer 21

i don't think your antiderivative is correct. you can check your antiderivative by differentiating it.

Answer 22

but the antiderivative ok e^k is (1/k)e^k
so wouldnt e^-x be (-1/x)e^-x
so when x is 8 it is (-1/8)e^-8
if i find the derivative of that is it e^-8

Answer 23

which is the format of the variable in the original equation correct?

Answer 24

the antiderivative of \[\huge \int\limits_{}^{} e^{\color{red} kx}dx=\frac{ 1 }{ k }e^{\color{red}k x}\]
in your f(x), k = -1.

Answer 25

so it would be -e^-x+x"?

Answer 26

so the exact answer is ((-e^-8)+8)-((-e^0)+0)
which is ((-e^-8)+8)-(-1)?

Answer 27

yup! its right :)
thx