Question:

Let f(x) be a continuous function on the interval [0, 5] defined by the equation f(x) = 2x + 3. Find the exact value of the definite integral ∫[2, 3] f(x) dx and interpret its meaning in the context of the function.

Answer:

To find the definite integral ∫[2, 3] f(x) dx, we will utilize the Fundamental Theorem of Calculus. According to the Fundamental Theorem of Calculus, if F(x) is an antiderivative of f(x) on an interval [a, b], then:

∫[a, b] f(x) dx = F(b) - F(a)

First, we need to find the antiderivative F(x) of the given function f(x) = 2x + 3. This is obtained by taking the integral with respect to x:

F(x) = ∫(2x + 3) dx = x^2 + 3x + C

Where C is the constant of integration.

Now, we need to evaluate F(3) and F(2) and subtract the values to find the definite integral ∫[2, 3] f(x) dx:

∫[2, 3] f(x) dx = F(3) - F(2) = (3)^2 + 3(3) - [(2)^2 + 3(2)] = 9 + 9 - (4 + 6) = 18 - 10 = 8

The exact value of the definite integral ∫[2, 3] f(x) dx is 8.

Interpretation: In the context of the function f(x) = 2x + 3 and the interval [2, 3], the value of the definite integral represents the net area bounded by the graph of the function and the x-axis over the given interval. In this case, the value 8 represents the net area bounded by the graph of the function and the x-axis between x = 2 and x = 3.