Question

Evaluate: \(\displaystyle \int_0^4 \sqrt{x} \, dx = \)

• A. \(2\)
• B. \(\frac{6}{\pi}\)
• C. \(\frac{4}{\pi}\)
• D. \(\frac{2}{\pi}\)

Hint:

For integrals of the form \(\int x^n dx\), use the power rule carefully, and calculate powers accurately.

Answer 1

The Correct Option is A

Step 1: Rewrite the integral: \[ \int_0^4 \sqrt{x} \, dx = \int_0^4 x^{\frac{1}{2}} \, dx \] Step 2: Integrate using the power rule: \[ \int x^{n} dx = \frac{x^{n+1}}{n+1} + c, \quad n \neq -1 \] \[ \implies \int_0^4 x^{\frac{1}{2}} \, dx = \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right]_0^4 = \frac{2}{3} \left[ x^{\frac{3}{2}} \right]_0^4 \] Step 3: Calculate \(4^{\frac{3}{2}}\): \[ 4^{\frac{3}{2}} = ( \sqrt{4} )^{3} = 2^{3} = 8 \] Step 4: Substitute: \[ \frac{2}{3} (8 - 0) = \frac{16}{3} \] None of the options match this answer, so please verify the question or options. ---