Calculus How To

Finding Area by the Limit Definition

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Finding Area by the Limit Definition

The area of a plane region (i.e., a 2-D shape) between the graph of a function and the x-axis can be found using the limit definition:
finding area with the limit definition

The formula looks complicated, but all you have to do is substitute in your values for the function ( f ) and interval ( ba ). From there, you can simplify and solve.

Finding Area by the Limit Definition: Example

Example question: Find the area (using the limit definition) between f(x) = 2x3 – x2 and the x-axis, on the interval [1, 2].

Step 1: (Optional) Make a graph. This not only gives you a visual of what you are trying to find, but you can ballpark your solution to see if your calculated answer makes sense.
finding area with the limit definition graph

Step 2: Find f (ci) by substituting in your interval (a and b) values into the f(ci) formula given above. We are finding the area for the interval between 1 and 2, so substituting those in and simplifying gives us:
step 2 area with limits

Step 3: Find Δx by substituting your a and b values into the Δx formula from above:
step 3 area with limit definition

Step 4: Substitute your answers from Step 2 and Step 3 into the main limit definition formula from above:
area with limits step 4

Step 5: Change the generic “f(x)” part of the formula (from Step 4) to your actual function and simplify. Our function for this example is f(x) = 2x3 – x2, so:
substituting in the function

Multiplying (1/n) through each term:
multiplying through term

Step 6: Distribute the summation symbol through all terms in your answer from Step 5:

Step 7: Apply summation properties and simplify.

Summation properties:

Term Summation

Step 8: Find the limit of all the “parts” from Step 7. For this example, we need one property of limits, which is the limit (1/x) = 0 (as x→∞) is 0:
Area = 0 + 0 + 1 + (1/2) + (5/3) + 2 = (31/6).

The solution is 31/6 ≈ 5.167.


Graph created with
Question from Calculus of a Single Variable, 10th edition. 4.2 Exercises, page 264.

Stephanie Glen. "Finding Area by the Limit Definition" From Calculus for the rest of us!

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