Area bounded by graph of function

Always remember that learning this is essential to the understanding of mathematics and will further your knowledge if so. Practicing makes perfect.


Here are some exercises to practice the topic of Area bounded by graph of a function. For this, it is important that you know the formulas and concepts of Volume of Revolution of solids. Last post is on the topic and you can review it by practicing the problems given.


Here are  2 examples for area bounded by equations on a graph or diagram.

Area bounded by equations #1 question.jpg

Question #1. And factorization of given equations to get the limits for the solution of the exercise. Final answer is also given as A=64/3.

Area bounded by equations #1 solution.jpg

Development of the problem.  We are following the formula f(x)-g(x) dx. Then we get the antiderivatives, and after we substitute into the x, first with 2 (The highest) which is f(x), and then g(x) as -2. From this we get that 32/3 – ( -32/3) and always remember these parenthesis because it is SUPER IMPORTANT to distribute the sign.

Exercise #2.


Question. And limits are solved for. Answer is also given. Try this exercise on your own!

You can also find more questions here –>  Practice Problems . Try a few! 🙂


Volume of Revolution of solids -#2 of 3 methods

ALWAYS REMEMBER: Your limits for the integral- substitute the higher value first.

This means, in the formula: V= integral {f(x)-g(x)} dx , when you’re substituting the x, you place the highest value first.


Method #2


The Washer Method description.


Use the Washer method to solve this question. Here, the limits are shown (-1,2) as well as the answer V=162/5 pi. Try it on your own 🙂


Check your process.

Unit Vector

Unit vectors are essential in mathematics.

A vector is a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.

The formula for unit vector is:

unit vector.png

Where the Unit Vector is equal to the vector divided by its magnitude. To do this, reasonably you will first need to solve for the magnitude. For instance, you will get your magnitude by writing the i,j,k values linearly squared, under a square root. As in the following example:


To understand the concept better, here is an example exercise.

Unit Vector Example 1.png

This is just an example with vector v. Oftentimes you will find 3D vectors with i,j,k values. Remember it’s always the same process. Just pay attention to the correct substitution and correct places where your values go.

A letter vector –> a vector.gif

Continue studying math, and the best wishes!🙂 🙂

Link to video on: “Finding a Unit Vector-Example”