Area bounded by a curve, the x-axis and a line perpendicular to the x-axis is called a quadrilateral. Quadrilaterals can be rectangles, squares, triangles or rhombuses depending on their shape.
Figuring out the area of a quadrilateral can be tricky at times. Luckily for you, we have done it before for both rectangles and squares! Read our previous articles on the area of quadrilaterals to help you out.
The first thing you need to do is figure out what kind of quadrilateral you are dealing with. This is easy if you break down the shape into its components.
Here are some helpful tips to identify different quadrilaterals and how to find their areas.
Find the area of the trapezoid T3
A trapezoid is a quadrilateral with no parallel sides. The base of the trapezoid is the set of points defined by the relationship between the two sides and the included diagonal.
The height of the trapezoid is the set of points defined by the relationship between the two non-parallel sides and the included bottom edge. The area of a trapezoid is found by dividing its total base length by its total top length.
This area formula can be used to find the area enclosed by any curve, provided that you can find these two values for your shape. You will need to find both the base length and the height for each side of your trapezoid, which can be tricky.
Use calculus to find the area
Once you have the boundaries of an area, you can find the area by dividing the space into thin strips, counting how many there are and how long they are, and then adding them all up.
Calculus uses a different approach: it considers the area to be a finite collection of infinitely thin strips, and calculates the average length of these strips to find the area.
Because calculus allows us to work with very small and very large numbers, it can also be used to find areas. For example, we can use calculus to find the average radius of a circle, which is its circumference. This is very useful in applications such as blood flow testing.
Find the length of the curve
To find the length of the curve, you need to calculate the radius of the circle that the curve passes through and multiply that by the number of revolutions it makes.
The more revolutions the curve makes, the longer it is. How many revolutions your curve makes depends on how many times it crosses the axis it curves around.
In this case, we have a semicircle, so it makes one revolution as it crosses the Y-axis once. The length of this curve is therefore its radius multiplied by 1, or .
The area of one half of this shape is square units. Multiply that by 2 to get , or square units total area.
Convert from polar to rectangular coordinates
The area enclosed by a curve and a straight line is called a strip. The length of the strip is calculated by finding the difference between the coordinates of the endpoints of the curve and the length of the straight line.
To convert from polar to rectangular coordinates, first find the radius r using the equation for distance, then use x = rθ and y = θ.
For instance, suppose you have the circle with radius 5, centered at (3, 2). Find the area enclosed by this circle and y = 2−x.
First, find r: r = 5. Then x = 2−0.5=1.5, so y = 1.5=2−x. The rectangle is (1.5, 2) × (2, x).
Find the area of rectangles that make up the curve
The first method for finding the area under a curve is to find the area of rectangles that make up the graph. This is done by breaking up the curve into a series of rectangles and finding the total area of these rectangles.
The height of each rectangle is found by taking the y-coordinate of the top of the rectangle and replacing t with t2-2t. The width of each rectangle is found by taking the length of one side of the rectangle and replacing t with t.
By adding up all of these smaller rectangles, you get the total area under this curve. This can be proven by doing a few examples. Try it yourself to see if you get similar results!
This method is not very accurate, however. It tends to over-estimate the area under the curve.
Sum all of the areas to find the total area
Another area problem that may give you trouble is finding the area enclosed by a curve, a line, and a plane. To do this, you will have to find the total area above the curve, below the curve, and on either side of the line.
Then you must add these areas together to get the total area enclosed by the curve, line, and plane. You can think of this as breaking down the area into little sub-areas and then adding them all up.
Try solving this problem by first drawing a sketch of the area so that you can see how all of these areas fit together. Next, find each individual area and lastly add them up to find the total area enclosed by the curve, line, and plane.
Leave a Reply