# Calculus 1 : Functions

## Example Questions

### Example Question #4110 : Calculus

To find area under a curve, you must:

integrate the function twice.

integrate the function once.

differentiate the function twice.

differentiate the function once.

integrate the function once.

Explanation:

To find the area of a curve, you are not finding the slope (thus not differentiating) but rather integrating. Specifically, you only differentiate once to find 2-dimensional area. Thus, the answer is "integrate the function once."

### Example Question #111 : How To Find Area Of A Region

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #111 : How To Find Area Of A Region

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #121 : Area

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #122 : Area

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #123 : Area

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #124 : Area

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #125 : Area

Find the area under the curve  between  and .

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

### Example Question #126 : Area

Find the area of the region bounded on top by  and on bottom by  from  to

Explanation:

To find the area bounded by the two curves, we must integrate:

, where f(x) is the top function and g(x) is the bottom function.

Using the above formula, we get

The integration was perfomed using the following rule:

The definite integration was performed by taking the result of the integral and evaluating it at the upper limit, and then subtracting the evaluation at the lower limit.

### Example Question #127 : Area

Write the function describing the area below  and above

Explanation:

To determine the function that describes the area between two functions, we must integrate over the difference between the upper and lower functions:

For ease, we can split the integral into three integrals:

For the first integral, we must make the following substitution:

Rewriting the integral and integrating, we get

which was found using the following rule:

The second integral is equal to

and was found using the following rule:

The third integral, found using the same rule as the first integral, is equal to

Combining the results - and adding all the constants of integration to make a single C - we get

.