Calculate the area of the shaded region between the 2 functions from $x=0$ to $x=9$

• A. 7
• B. 10
• C. 4.5
• D. 445

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the two functions that bound the shaded region. From the graph, the upper function is f(x) = √x and the lower function is g(x) = x/3.
2. Step 2: Set up the integral to calculate the area between the two curves. The area is given by the integral of the difference between the upper function and the lower function over the interval [0, 9]. This can be expressed as: ∫[0,9] (√x - x/3) dx.
3. Step 3: Break down the integral into two separate integrals for easier computation: ∫[0,9] √x dx - ∫[0,9] (x/3) dx.
4. Step 4: Compute the antiderivative of each term. For √x, rewrite it as x^(1/2) and use the power rule to find the antiderivative: (2/3)x^(3/2). For x/3, factor out 1/3 and use the power rule to find the antiderivative: (1/6)x^2.
5. Step 5: Evaluate the definite integrals by substituting the limits of integration (x = 0 and x = 9) into the antiderivatives. Subtract the results of the two integrals to find the total area of the shaded region.