Calculus - Integration - Finding areas.
Summary of the steps.
There are several types of question focused on finding area using calculus.
The most common formats for the functions we encounter - are listed below
together with the steps to follow to obtain the require area:
1. Basic area between the curve for a function and the x axis: | Step 1: | Draw a rough sketch of the function. |
Step 2: | Add a thin slice from the x-axis to the curve (i.e. the y distance). |
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Step 3: | Write the integral statement as to summarise the area of the slice. |
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Step 4: | Include the limits between which your slice moves - for example |
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Step 5: | Replace y with the function in x. | |
Step 6: | Evaluate the integral in x using the appropriate technique. | |
Step 7: | Report your result. | |
2. Basic area between the curve for a function and the y axis: | Step 1: | Draw a rough sketch of the function. |
Step 2: | Add a thin slice from the y-axis to the curve (i.e. the x distance). |
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Step 3: | Write the integral statement as to summarise the area of the slice. |
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Step 4: | Note the limits between which your slice moves (here in the vertical direction) - for example |
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Step 5: | Replace x with the function in y. | |
Step 6: | Evaluate the integral in y using the appropriate technique. | |
Step 7: | Report your result. | |
3. Area between 2 curves with one point of intersection. | Step 1: | Draw a rough sketch of the two functions. Calculate the x value for their point of intersection (if not given). |
Step 2: | Add two thin slices from the x-axis - one to each of the curves (i.e. the y distance). |
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Steps 3 - 6: | Follow steps 3 to 6 above for type 1 for each of the two areas separately. |
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Step 7: | Add the two separate areas together. | |
Area between 2 curves with two points of intersection. | Step 1: | Draw a rough sketch of the two functions. Calculate the x values for their points of intersection (if not given). |
Step 2: | Add a thin slice starting from the top curve and finishing at the lower curve (i.e. the y distance between the two functions). |
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Step 3: | Write the integral statement as to summarise the area of the slice with y being the distance between the curves |
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Step 4: | Note the limits between which your slice moves - for example here the two points of intersection are |
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Step 5: | Replace y (or f(x) and g(x)) with the difference between the functions in terms of x. NOTE: It does not matter if one of your curves goes below the x axis. The difference approach eliminates any problems. |
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Step 6: | Evaluate the integral in x using the appropriate technique. | |
Step 7: | Report your result. |