Find the average value of the function $$ f(x)=e^{x} $$ over the interval $$ [1,3] $$ $$ e^{[?]}- $$

Question

UpStudy AI · Accepted Answer

We start by recalling that the average value of a function $$ f(x) $$ on the interval $$[a,b]$$ is given by

$$
A = \frac{1}{b-a}\int_a^b f(x)\,dx.
$$

For the function $$ f(x)=e^x $$ on the interval $$[1,3]$$, we have

$$
A = \frac{1}{3-1}\int_1^3 e^x\,dx.
$$

This simplifies to

$$
A = \frac{1}{2}\int_1^3 e^x\,dx.
$$

Next, we compute the integral. The antiderivative of $$ e^x $$ is $$ e^x $$ itself, so

$$
\int_1^3 e^x\,dx = e^x \Big|_1^3 = e^3 - e.
$$

Substituting back into the formula for $$ A $$, we obtain

$$
A = \frac{1}{2}(e^3-e).
$$

Thus, the average value of $$ f(x)=e^x $$ over the interval $$[1,3]$$ is

$$
\boxed{\frac{e^3-e}{2}}.
$$
The average value of $$ f(x) = e^x $$ over the interval $$[1,3]$$ is $$ \frac{e^3 - e}{2} $$.

Find the average value of the
function

over the interval