Question

If f(x)=8x then what is the area enclosed by the graph of the function, the horizontal axis, and vertical lines at x=2 and x=6

Answer 1

Answer:

$A=128$

Step-by-step explanation:

First of all we need to graph f(x)=8x, (First picture)

Now we have to calculate the area enclosed by the graph of the function, the horizontal axis, and vertical lines at $x_{1}=2$ and $x_{2}=6$ ,

The area that we have to calculate is in pink (second picture).

The function is positive and the domain is $[2,6]$ then we can calculate the area with this formula:

$A=\int\limits^b_a {f(x)} \, dx$,

In this case $b=x_{2} , a=x_{1}$

$A=\int\limits^6_2 {8x} \, dx = 8\int\limits^6_2 {x} \, dx$

The result of the integral is,

$A=8\frac{x^{2}}{2}$, but the integral is defined in [2,6] so we have to apply Barrow's rule,

Barrow's rule:

If f is continuous in [a,b] and F is a primitive of f in [a,b], then:

$\int\limits^b_a {f(x)} \, dx =F(b)-F(a)$

Applying Barrow's rule the result is:

$A=8.\frac{6^{2} }{2}-8.\frac{2^{2} }{2}$

$A=8.\frac{36}{2} -8.\frac{4}{2}$

$A=144-16$

$A=128$