To find the area under the graph of f(x) = 2^(-3x) and above the x-axis (f(x) = 0) for x in the interval [1, 7], we can use integration. By integrating the function over this interval, we can calculate the area. Let's proceed with the integration:
∫[1,7] 2^(-3x) dx
To evaluate this integral, we can use the formula for the integral of a power function:
∫ x^n dx = (1/(n+1)) * x^(n+1)
Applying this formula, we have:
∫[1,7] 2^(-3x) dx = (1/(-3)) * 2^(-3x+1) evaluated from 1 to 7
Now let's substitute the values and calculate the area:
(1/(-3)) * (2^(-3*7+1) - 2^(-3*1+1))
Simplifying further:
(1/(-3)) * (2^(-20) - 2^(-2))
Calculating this expression will give us the area under the graph of f(x) = 2^(-3x) and above the x-axis for x in [1, 7].
