Evaluate the integral by interpreting it in terms of areas. integral -3 to 0(1+(9-x^2)^1/2)dx

Step-by-step explanation:

We need to evaluate the following integral by interpreting it in terms of areas :

The first step is to separate the integral into two easier integrals

   (Integral of the sum)

Now we can calculate each integral by studying the area below each function.

For the first integral the function is

(I will attach a file with the functions)

The area below this function is the area of a rectangle with sides 1 and 3 ⇒

For the second integral the function is

If we study this function :

(I)

Which is the equation of a circle centered at (0,0) with radius equal to 3

From the equation (I)

The two possible solutions are :

(II) and

We will use (II) to solve the integral (which is the upper part of the circle)

The area of a circle with radius equal to 3 is

In the integral we only need a quarter of circle ⇒ We divide the total area by 4 ⇒ ⇒

Finally the integral is equal to

y/x=18

step-by-step explanation:

you can first factor out the n to get n(1+5). we know that 1+5 is 6, so we just have 6n.