Questions: Find the definite integral ∫ from 4 to 9 of 3√x dx.

Solution Steps

To find the definite integral \( \int_{4}^{9} 3 \sqrt{x} \, dx \), we first determine the antiderivative of the function \( 3 \sqrt{x} \). The antiderivative is given by: \[ \int 3 \sqrt{x} \, dx = 3 \cdot \frac{2}{3} x^{3/2} = 2 x^{3/2} \]

Next, we evaluate the antiderivative at the upper limit \( x = 9 \) and the lower limit \( x = 4 \): \[ F(9) = 2 \cdot 9^{3/2} = 2 \cdot 27 = 54 \] \[ F(4) = 2 \cdot 4^{3/2} = 2 \cdot 8 = 16 \]

Now, we subtract the value of the antiderivative at the lower limit from the value at the upper limit: \[ \int_{4}^{9} 3 \sqrt{x} \, dx = F(9) - F(4) = 54 - 16 = 38 \]

Final Answer