91Ó°ÊÓ

We need to find the antiderivative of the function, $$f(x) = x - \sqrt{x}$$. To do this, rewrite the integrand and then find the antiderivative for each term. Rewrite \(\sqrt{x}\) as \(x^{\frac{1}{2}}\). The function becomes: $$f(x) = x - x^{\frac{1}{2}}$$ Now, we find the antiderivative of each term individually: $$\int x \: dx = \frac{1}{2}x^{2}$$ $$\int x^{\frac{1}{2}} \: dx = \frac{2}{3}x^{\frac{3}{2}}$$ Now that we have the antiderivatives of each term, we can add them together to get the antiderivative of the function: $$F(x) = \frac{1}{2}x^{2} - \frac{2}{3}x^{\frac{3}{2}}$$

Now that we have found the antiderivative for the function $$f(x) = x - x^{\frac{1}{2}}$$, we can use the Fundamental Theorem of Calculus to evaluate the definite integral. The theorem states that for a continuous function f(x) on an interval [a, b], if F(x) is an antiderivative of f(x), then: $$\int_{a}^{b} f(x) \: dx = F(b) - F(a)$$ In this problem, we are given the limits of integration as a = 0 and b = 1. Use these values along with the antiderivative function we found in step 1: $$\int_{0}^{1}(x - x^{\frac{1}{2}}) \: dx = \left(\frac{1}{â‚‚}(1)^{2} - \frac{2}{3}(1)^{\frac{3}{2}}\right) - \left(\frac{1}{2}(0)^{2} - \frac{2}{3}(0)^{\frac{3}{2}}\right)$$ After calculating the values, we get: $$\int_{0}^{1}(x - x^{\frac{1}{2}}) \: dx = \left(\frac{1}{2} - \frac{2}{3}\right) - \left(0 - 0\right) = \frac{1}{6}$$

Now that we have evaluated the integral, we need to sketch the graph of the function $$f(x) = x - x^{\frac{1}{2}}$$ on the interval [0, 1]. The function has an increasing portion on the interval, starting from the origin (0,0) and ending at the point (1,1-1)= (1,0), crossing the x-axis at that point. To shade the region representing the net area we have found, draw a vertical line at x=1 and notice the enclosed region between the function and the x-axis on the interval [0, 1]. This region represents the definite integral we found, which has a value of \(\frac{1}{6}\). In conclusion, we have found that the definite integral of the function $$f(x) = x - x^{\frac{1}{2}}$$ on the interval [0, 1] is \(\frac{1}{6}\), and we have sketched the graph of the function along with the region representing the net area we found.