Question

What is the largest value that∫ from a to b x√(2x - x²) dxcan have for any a and b? Give reasons for your answer.

Accepted Answer

First, understand the problem: we want to find the largest possible value of the definite integral $$\int_a^b x \sqrt{2x - x^2$} \, dx$$ for any choice of limits $$a$$ and $$b. $$This means we are looking for the maximum area under the curve of the function $$f(x) = x \sqrt{2x - x^2$}$$ over some interval $$[a,b]$$ within the domain where the integrand is defined and real. Determine the domain of the integrand $$f(x) = x \sqrt{2x - x^2$}. $$Since the expression inside the square root must be non-negative, solve $$2x - x^2$ \geq 0. $$Factor this as $$x(2 - x) \geq 0$$, which implies $$x \in [0, 2]. So $$the function is real-valued and defined on the interval $$[0, 2]. To $$find the largest value of the integral, consider that the integral over any subinterval $$[a,b] \subseteq [0,2]$$ represents the area under the curve $$f(x). $$The maximum integral value will be the integral over the interval where the function is positive and the area is largest. Since the function is zero at the endpoints $$x=0$$ and $$x=2$$, and positive in between, the maximum integral is likely over the entire interval $$[0,2]. $$Set up the integral over the full domain: $$I = \$$int_0$$^2 x \sqrt{2x - x^2$} \, dx. To $$solve this integral, use an appropriate substitution. For example, let $$u = 2x - x^2$, then find du$ in terms of dx$ and express x$ in terms of u$ to rewrite the integral in terms of u$. Alternatively, consider a trigonometric substitution to simplify the square root. After substitution, rewrite the integral in a simpler form and evaluate it (this step involves integration techniques such as substitution or trigonometric substitution). The value of this integral over $$[0,2]$$ will give the largest possible value of the original integral for any a$ and b$ because any smaller interval will yield a smaller area.

What is the largest value that
∫ from a to b x√(2x - x²) dx
can have for any a and b? Give reasons for your answer.

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Key Concepts

Definite Integral and Area Interpretation

Domain and Behavior of the Integrand

Maximizing the Integral by Choosing Limits