Question

find the area between the graph of $f(x) = 7sqrt{x}$ and the $x$-axis on the interval $4, 9$. write the exact answer. do not round.

Explanation:

Step1: Rewrite function for integration

Rewrite $7\sqrt{x}$ as $7x^{\frac{1}{2}}$

Step2: Apply power rule for integration

The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$. For $n=\frac{1}{2}$:
$$\int 7x^{\frac{1}{2}}dx = 7 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{14}{3}x^{\frac{3}{2}} + C$$

Step3: Evaluate definite integral on [4,9]

Use the Fundamental Theorem of Calculus: $\int_{a}^{b} f(x)dx = F(b)-F(a)$
$$\frac{14}{3}(9)^{\frac{3}{2}} - \frac{14}{3}(4)^{\frac{3}{2}}$$

Step4: Simplify the terms

Calculate $9^{\frac{3}{2}}=(9^{\frac{1}{2}})^3=3^3=27$, $4^{\frac{3}{2}}=(4^{\frac{1}{2}})^3=2^3=8$:
$$\frac{14}{3}(27) - \frac{14}{3}(8) = 14 \cdot 9 - \frac{112}{3} = 126 - \frac{112}{3}$$

Step5: Compute final subtraction

Convert to common denominator:
$$\frac{378}{3} - \frac{112}{3} = \frac{266}{3}$$

Answer:

$\frac{266}{3}$