Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #131 : How To Find Integral Expressions

Solve the definite integral.

Possible Answers:

None of these

Correct answer:

Explanation:

To solve this integral we must first realize that an integral is the opposite operation of a derivative and that the derivative of  is .

This means that  is our answer where A is a constant. We must take the chain rule into account to figure out the constant associated with our answer.

The derivative of  is  so that means we must multiply our integral by  to get our answer.

Also, we can't forget about the original 2 in the numerator.

Our integral then becomes 

.

To solve a definite integral we must take the difference of the function at the upper and lower limits. The answer is then

.

By the logarithmic divison rules, this answer is

.

Example Question #132 : How To Find Integral Expressions

Solve the following problem.

Possible Answers:

Correct answer:

Explanation:

Because we know that , this problem becomes easier than it looks.  To solve this, we actually use something known as the u-substitution method.  By setting , we are one step closer to solving this.  After deriving both sides with respect to , we have .

By multiplying both sides by , we have 

 This allows us to be able to rewrite the problem as 

, giving us our final answer.

Example Question #133 : How To Find Integral Expressions

Find the average value of the following function over the given interval.

 over the interval 

Possible Answers:

Correct answer:

Explanation:

To find the average value of this function, we must find the total value of the area under the curve, and divide it by the number that represents the range of our interval , which is .  We can then use our understanding of integral calculus to find the anti-derivative and use the end points to find the total area under the curve.

 

 

 

If we divide this number by , we will come out with our average value, .

Example Question #134 : How To Find Integral Expressions

Compute the following integral.

Possible Answers:

Correct answer:

Explanation:

The only difference between this problem and a normal integral is the absolute value, which makes a small change in the computation and analytical understanding of the problem.  Graphically, the part of the function that would normally be under the x-axis is now flipped and positive.  This only means that we need to break this integral up into at least two parts.

First, we need to find the roots of our function. Because our function can be factored like this , we can see that our roots will be .  If we quickly analyze the graph, we can see that any value inside the interval  would normally be negative while values outside of this interval would normally be positive.  In order to solve the problem, we need to change all values of our function that are inside the given interval to positive values.

Once we have done this, we can add up the area of these three sections  (by using anti-derivatives) and come up with the answer of .

Example Question #135 : How To Find Integral Expressions

Which of the following integral expressions could be used to find the area bound by , the -axis, and the line .

Possible Answers:

Correct answer:

Explanation:

Which of the following integral expressions could be used to find the area bound by g(t), the y-axis, and the line .

The question asks us to set up an integral to find the area of a given region. Because we are given limits of integration, we must set up a definite integral (one with numbers at the top and bottom of the integral sign). 

Furthermore, we want our lower limit to be 0 and our upper limit to be 5. We know this because we need the area from the y-axis to 5.

So our integral should start like this:

Next, we want to put the function we are given within the integral as well, making the anwser:

Example Question #136 : How To Find Integral Expressions

Which of the following is the correct integral expression for this statement:

Find the area of the region bound by , the  and  axes, and the line .

Possible Answers:

Correct answer:

Explanation:

Which of the following is the correct integral expression for this statement: Find the area of the region bound by f(z), the y and z axes, and the line .

Let's begin here by not getting too worried. Note that although this function may look intimidating, we don't actually have to calculate anything. All we are going to do is set this puppy up.

The first step is to set up the correct limits of integration. We want the smallest of the two to go on bottom. 

Now, you may be used to always having zero on the bottom, but -6 is smaller, so we will put -6 down below.

Next, it's just a matter of putting in f(z) and we have our integral:

Example Question #137 : Integral Expressions

Find the integral expression of area of a circle  with respect to its circumference  and radius 

Possible Answers:

Correct answer:

Explanation:

Area of a circle is given by 

 and its circumference 

Notice how circumference is the derivative of area with respect to radius 

Using the power rule we know that:

, where and are constants.

 

Writing this as an integral,

Example Question #138 : Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

The integral is equal to 

and was found using the following rules:

Example Question #137 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral we must perform the following substitution:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with our original term:

Example Question #138 : How To Find Integral Expressions

Evaluate Integral

Evaluate,

Possible Answers:

None of the above

Correct answer:

Explanation:

Compute the indefinite integral,

Compute the boundaries,

Simplify,

 

 

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