Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #4111 : Calculus

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

 

With our final answer being:

Example Question #4112 : Calculus

Find the area under the curve  between  and .

 

 
Possible Answers:

Correct answer:

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:

With our final answer being:

Example Question #201 : Regions

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

 

With our final answer being:

Example Question #4114 : Calculus

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

With our final answer being:

Example Question #4115 : Calculus

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

With our final answer being:

Example Question #121 : How To Find Area Of A Region

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

With our final answer being:

Example Question #4117 : Calculus

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

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:

 

With our final answer being:

Example Question #121 : How To Find Area Of A Region

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

Possible Answers:

Correct answer:

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 #4119 : Calculus

Write the function describing the area below  and above 

Possible Answers:

Correct answer:

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

.

Example Question #122 : How To Find Area Of A Region

Find the area of the region bound by the lines , the x-axis, and the function v(x)

Possible Answers:

Correct answer:

Explanation:

Find the area of the region bound by the lines , the x-axis, and the function v(x)

First, we need to set up our integral.

It should look like this:

Next, recall the rule for integrating polynomials: Increase each exponent by 1 and divide by that number:

Simplify this to get

Next, evaluate our integral by finding the difference between V(10) and V(5)

So our answer is 19939770

 

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