Calculate the area under the curve y=e^-x above the x axis on the interval [1, infinity).
a. infinity
b. 0
c. -1/e
d. 1/e
e. none of the above
integral e^-x dx = -e^-x
so
-e^-oo - (-e^-1)
= 0 + 1/e
To calculate the area under the curve y=e^-x above the x axis on the interval [1, infinity), we need to integrate the function over that interval.
The integral of y=e^-x with respect to x on the given interval can be expressed as:
∫[1,∞] e^-x dx
To solve this integral, we can use the fact that ∫ e^-x dx = -e^-x + C, where C is the constant of integration.
Therefore, the integral becomes:
∫[1,∞] e^-x dx = [-e^-x] evaluated from 1 to ∞
Evaluating the limits of the integral:
lim(∞->∞) -e^-∞ - (-e^-1)
Since e^-∞ approaches 0 and e^-1 is a positive constant, we can simplify the equation to:
0 - (-e^-1) = e^-1 = 1/e
Therefore, the area under the curve y=e^-x above the x axis on the interval [1, infinity) is equal to 1/e.
Answer: d. 1/e
To calculate the area under the curve y=e^(-x) above the x-axis on the interval [1, infinity), we can use definite integration.
The definite integral of a function f(x) from a to b represents the area under the curve of the function between x=a and x=b. In this case, we want to find the area under the curve from x=1 to infinity, so our integral will be:
∫(1 to infinity) e^(-x) dx
To evaluate this integral, we can use the properties of the exponential function and the rules of integration. The integral of e^(-x) is -e^(-x).
Applying these rules, we get:
= -e^(-x) |(1 to infinity)
= -[e^(-infinity) - e^(-1)]
Now, as x approaches infinity, e^(-infinity) approaches 0 since the exponential function decreases rapidly as x gets larger. So our expression simplifies to:
= -[0 - e^(-1)]
= -[-e^(-1)]
= e^(-1)
= 1/e
Therefore, the area under the curve y=e^(-x) above the x-axis on the interval [1, infinity) is 1/e.
So the correct answer is:
d. 1/e