Question

При якому значенні параметра а пряма х=а ділить площу фігури, обмеженої графіком функції f(x)=8/x і прямими у=0, х=2, х=8 навпіл?

Answer 1

Ответ:

a=4

x=4

Объяснение:

∫(8/x)dx=8lnx

8ln8-8ln2=8*3ln2-8ln2=16ln2

8lna-8ln2=8ln2

lna=2ln2

lna=ln4

a=4

Answer 2

Finding the Value of Parameter *a* for which the Line *x=a* Divides the Area of the Region Bounded by the Graph of *f(x)=8/x* and the Lines *y=0*, *x=2*, and *x=8* in Half

To find the value of the parameter *a* for which the line *x=a* divides the area of the region bounded by the graph of *f(x)=8/x* and the lines *y=0*, *x=2*, and *x=8* in half, we can use the concept of definite integrals to calculate the area of the region and then determine the value of *a*.

Calculating the Area of the Bounded Region

The area of the region bounded by the graph of *f(x)=8/x* and the lines *y=0*, *x=2*, and *x=8* can be calculated by finding the definite integral of the function *f(x)=8/x* over the interval [2, 8]. The definite integral of a function *f(x)* over the interval [a, b] represents the signed area between the *x*-axis and the curve of the function between the limits of integration.

The definite integral to calculate the area of the bounded region is given by:

A = ∫[2, 8] 8/x dx

Let's calculate this definite integral to find the area *A* of the bounded region.

Finding the Value of *a* that Divides the Area in Half

Once we have the value of the area *A*, we can then find the value of the parameter *a* for which the line *x=a* divides this area in half. This can be done by setting up and solving an equation based on the properties of definite integrals and the area of the bounded region.

I will now calculate the definite integral to find the area of the bounded region and then proceed to find the value of *a* that divides this area in half.