To find the area under the curve

Integrate it from the specified bounds:

The first step is determine the lower and upper x-values that define the area. There is a lower bound of zero that marks the transition for f(x) to move into negative y-values; however, g(x) is well into the negative at this point, so it'll be necessary to find a lower bound where it first begins to become negative. This will occur for a value of five:

This allows the creation of an initial integral:

Another upper bound can be found by determining the point where the two functions intersect:

Now, integrate the difference of these functions over these final bounds:

The full area is now the sum of these two:

In order to find the area under  on the interval of  to , you must evaluate the definite integral

First, antidifferentiate the function.

Then, substitute values for .

Finally, evaluate in terms of 

In order to find the area under  on the interval of  to , you must evaluate the definite integral

First, integrate the function.

Then, substitute values for .

Finally, evaluate while cancelling negative signs.

The area described is the integral 

.

This can be evaluated using the linear properties of integrals and the Power Law, which says

.

Thus the integral above is equal to

.

To the find the area of the region between  and , one needs to first figure out which function is above the other in the interval . Plug in  and  into each function to check which one has the higher value at these respective points.

 has a higher value than at both of these x-values, and since both functions are increasing during this interval, we can conclude that  is above  during this interval.

After that, set the definite integral 

.

Using the general rule for integrals,

and for exponential functions,

on our function we get

  from  to .

This equals to 

.

Area of a region between two graphs  and , and between two  values  and  can be given as

Since we want to find the area between  and  between  and ,

we can set this up as one definite integral. 

We can ignore the value until we have a numerical answer.

First, we know that by the sum rule

By the power rule, we know that

, where are constants and  is a variable.

Therefore,

We also know as an identity that

, when 

Therefore,

We also know that for definite integrals,

, where 

In our case, 

In order to find the area of the region between the curve of the function and the x-axis on the interval [,], we solve for the integral

.

For this problem the integral becomes

And since the fuction is always positive on the interval, the integral becomes

.

When taking the integral, we will use the inverse power rule which states,

 .

Applying this rule we get

 .

And by the corollary of the First Fundamental Theorem of Calculus

 .

As such, the area is 

 square units.

In order to find the area of the region between the curve of the function and the x-axis on the interval , we solve for the integral

.

For this problem the integral becomes

.

And since the function is always positive on the interval, the integral becomes

.

When taking the integral, we will use the inverse power rule which states,

 .

Applying this rule we get

 .

And by the corollary of the First Fundamental Theorem of Calculus

 .

As such, the area is 

 square units.

For the region to the right of the y-axis, the range of x is .  The next step is to find the integral of the function:

The method of integration by parts will serve here:

Now,  converges to zero; however, to determine what  conveges to, use L'Hopital's Rule:

Since  can be rewrittena s  and both the numerator and denominator approach  as x does: