Question

please helpFind the area between the graph of f(x) = ____ and the x-axis on the interval [4, 9]. Write the exact answer. Do not round.

Answer 1

Recall that the area between the graphs of two functions on the interval [a,b] is:

`\int ^b_a|f(x)-h(x)|dx\text{.}`

Therefore, the area between f(x)=4√x and the x-axis with equation y=0, on the interval [4,9] is:

`\int ^9_4|4\sqrt[]{x}-0|dx=\int ^9_4|4\sqrt[]{x}|dx\text{.}`

Notice that

`4\sqrt[]{x}>0,`

for all x in the given interval, therefore:

`\int ^9_4|4\sqrt[]{x}|dx=\int ^9_44\sqrt[]{x}dx=4\int ^9_4x^{(1)/(2)}dx=4(\frac{2x^{(3)/(2)}}{3})|^9_4=(8)/(3)(9^{(3)/(2)}-4^{(3)/(2)})=(8)/(3)(19)=(152)/(3).`

Answer: 152/3.