Visualizing the Definite Integral
From \( x = 2 \) to \( x = 5 \), under \( f(x) = \sqrt{x} \)
To find the total area under \( f(x) = \sqrt{x} \) from \( x = 2 \) to \( x = 5 \), we use the Fundamental Theorem of Calculus.
The antiderivative is \( F(x) = \frac{2}{3}x^{3/2} \), so \( F(5) - F(2) \) gives the area.
Steps for Finding the Antiderivative
Step 1: Rewrite the function with a rational exponent:
\( \sqrt{x} = x^{1/2} \)
Step 2: Identify \( a = 1 \), \( n = 1/2 \)
Step 3: Apply the formula:
\( \int x^n dx = \frac{a}{n+1} x^{n+1} + C \)
\( \int \sqrt{x} dx = \int x^{1/2} dx = \frac{2}{3}x^{3/2} + C \)