The average value of a function f(x) on the interval [a, b] is given by the formula:

1/(b - a) * ∫ from a to b of f(x) dx

In this case, f(x) = sin(π/2 * x), a = 2, and b = 4. So we need to compute:

1/(4 - 2) * ∫ from 2 to 4 of sin(π/2 * x) dx

This simplifies to:

1/2 * ∫ from 2 to 4 of sin(π/2 * x) dx

To compute the integral, we use the antiderivative of sin(π/2 * x), which is -2/π * cos(π/2 * x). Evaluating this from 2 to 4 gives:

1/2 * [-2/π * cos(π/2 * 4) - (-2/π * cos(π/2 * 2))]

This simplifies to:

1/2 * [-2/π * 1 - (-2/π * 0)]

Which simplifies further to:

-1/π

So, the average value of f on the interval [2, 4] is -1/π.

Question

The average value of a function f(x) on the interval [a, b] is given by the formula:

1/(b - a) * ∫ from a to b of f(x) dx

In this case, f(x) = sin(π/2 * x), a = 2, and b = 4. So we need to compute:

1/(4 - 2) * ∫ from 2 to 4 of sin(π/2 * x) dx

This simplifies to:

1/2 * ∫ from 2 to 4 of sin(π/2 * x) dx

To compute the integral, we use the antiderivative of sin(π/2 * x), which is -2/π * cos(π/2 * x). Evaluating this from 2 to 4 gives:

1/2 * [-2/π * cos(π/2 * 4) - (-2/π * cos(π/2 * 2))]

This simplifies to:

1/2 * [-2/π * 1 - (-2/π * 0)]

Which simplifies further to:

-1/π

So, the average value of f on the interval [2, 4] is -1/π.

Knowee AI · Accepted Answer

The average value of a function f(x) on the interval [a, b] is given by the formula:

1/(b - a) * ∫ from a to b of f(x) dx

In this case, f(x) = sin(π/2 * x), a = 2, and b = 4. So we need to compute:

1/(4 - 2) * ∫ from 2 to 4 of sin(π/2 * x) dx

This simplifies to:

1/2 * ∫ from 2 to 4 of sin(π/2 * x) dx

To compute the integral, we use the antiderivative of sin(π/2 * x), which is -2/π * cos(π/2 * x). Evaluating this from 2 to 4 gives:

1/2 * [-2/π * cos(π/2 * 4) - (-2/π * cos(π/2 * 2))]

This simplifies to:

1/2 * [-2/π * 1 - (-2/π * 0)]

Which simplifies further to:

-1/π

So, the average value of f on the interval [2, 4] is -1/π.

Let ff be the function defined by f, of, x, equals, sine, left bracket, start fraction, pi, divided by, 2, end fraction, x, right bracket, .f(x)=sin( 2π x). What is the average value of ff on the interval open square bracket, 2, comma, 4, close square bracket[2,4] written in simplest form?