Calculus 1 : How to find area of a region

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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Example Question #1 : How To Find Area Of A Region

What is the average value of the function f(x) = 12x3 + 15x + 5 on the interval [3, 6]?

Possible Answers:

1350.2

895.67

1542

1302.5

771

Correct answer:

1302.5

Explanation:

To find the average value, we must take the integral of f(x) between 3 and 6 and then multiply it by 1/(6 – 3) = 1/3.

The indefinite form of the integral is: 3x4 + 7.5x2 + 5x

The integral from 3 to 6 is therefore: (3(6)4 + 7.5(6)2 + 5(6)) - (3(3)4 + 7.5(3)2 + 5(3)) = (3888 + 270 + 30) – (243 + 22.5 + 15) = 3907.5

The average value is 3907.5/3 = 1302.5

Example Question #2 : How To Find Area Of A Region

Find the dot product of a = <2,2,-1> and b = <5,-3,2>.

Possible Answers:
2
-2
0
3
-3
Correct answer: 2
Explanation:

To find the dot product, we multiply the individual corresponding components and add.  

Here, the dot product is found by:

2 * 5 + 2 * (-3) + (-1) * 2 = 2.

Example Question #3 : How To Find Area Of A Region

Find the area of the region enclosed by the parabola  and the line 

Possible Answers:

Correct answer:

Explanation:

The limits of the integration are found by solving   and  for :





The region runs from  to . The limits of the integration are ,
The area between the curves is:



Example Question #4 : How To Find Area Of A Region

Find 

Possible Answers:

Correct answer:

Explanation:

Example Question #5 : How To Find Area Of A Region

Find 

Possible Answers:

Correct answer:

Explanation:

Example Question #6 : How To Find Area Of A Region

What is the area of the space below  and above 

Possible Answers:

Correct answer:

Explanation:

 is only above  over the interval .  Areas are given by the definite integral of each function and 

 

The area between the curves is found by subtracting the area between each curve and the -axis from each other.  For  this area is  and for  the area is  giving an area between curves of 

Example Question #7 : How To Find Area Of A Region

What is the area below  and above the -axis?

Possible Answers:

Correct answer:

Explanation:

To find the area below a curve, you must find the definite integral of the function.  In this case the limits of integration are where the original function intercepts the -axis at  and .  So you must find  which is  evaluated from  to .  This gives an answer of 

Example Question #8 : How To Find Area Of A Region

Find the area between the curves  and .

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we first need to find the point where the two equations are equal.  Doing this we find that

 .  

From this, we see that the two graphs are equal at  and .  We also know that for is greater than .  

So to find the area between these curves we need to evaluate the integral .  

The solution to the integral is

.

Evaluating this at  and  we get

Example Question #9 : How To Find Area Of A Region

Find the value of 

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we will need to do a -substitution.  Letting

 .  

Substituting our  function back into the integral, we get

 

Evaluating this at  and  we get

 

Example Question #10 : How To Find Area Of A Region

Find the average value of  on the interval

Possible Answers:

Correct answer:

Explanation:

The average is given by integration as:

This means that:

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