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AP Calculus AB: 10.2.3 Common Mistakes to Avoid When Finding Areas

Mathematics6 CardsCreated 3 months ago

This section highlights common errors students make when calculating areas using definite integrals. It emphasizes the importance of checking whether curves cross, breaking integrals when needed, and ensuring areas below the x-axis are treated as positive. Sketching the region helps prevent sign and setup mistakes.

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Common Mistakes to Avoid When Finding Areas

  • When evaluating areas of regions, make sure that the curves do not cross within the corresponding open interval. If they do, it is necessary to evaluate the area of each region separately.

  • When finding the area between the x-axis and the curve of a function underneath the axis, multiply the function by –1 to avoid getting a negative area.

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Key Terms

Term
Definition

Common Mistakes to Avoid When Finding Areas

  • When evaluating areas of regions, make sure that the curves do not cross within the corresponding open interval. If they do, it is necessar...

note

  • If the curves intersect, it is sometimes necessary to break the integral into two parts.

  • Anytime the way that the arbitrary ...

What is the area bound between the curve f (x) = x^ 2 − 4 and the x‑axis?

A = 32/3

Find the area of the region bound by y = 2x/3 + 3, y = 2, x = −3, x = +3.

7.5

Find the area of the region bound by y=x^3/2 and y=2x.

4

Find the area of the region bound by y = 4x/5, x = −3, x = +3, and the x-axis.

36/5

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AP Calculus AB: 10.2.3 Common Mistakes to Avoid When Finding Areas
TermDefinition

Common Mistakes to Avoid When Finding Areas

  • When evaluating areas of regions, make sure that the curves do not cross within the corresponding open interval. If they do, it is necessary to evaluate the area of each region separately.

  • When finding the area between the x-axis and the curve of a function underneath the axis, multiply the function by –1 to avoid getting a negative area.

note

  • If the curves intersect, it is sometimes necessary to break the integral into two parts.

  • Anytime the way that the arbitrary rectangle’s dimensions are defined changes, you will have to break the problem up into another piece.

  • Sometimes you can use the fact that the regions are similar to reduce the extra work you have to do.

  • A common mistake in finding areas is to assume that the area is equal to the definite integral and work without a picture.

  • The formula for finding area requires the way you define the arbitrary rectangles to be positive. Area cannot be negative.

  • To avoid this mistake, always consider how the arbitrary rectangle is defined. Notice that finding the area underneath the x-axis, but above a curve, requires that you put a negative sign into the definite integral.

  • Definite integrals can be negative, but areas cannot. If you get a negative answer when finding area, check how you defined the dimensions. You might have made an algebraic mistake, but it is more likely that you did not consider the fact that the area was below the x-axis.

What is the area bound between the curve f (x) = x^ 2 − 4 and the x‑axis?

A = 32/3

Find the area of the region bound by y = 2x/3 + 3, y = 2, x = −3, x = +3.

7.5

Find the area of the region bound by y=x^3/2 and y=2x.

4

Find the area of the region bound by y = 4x/5, x = −3, x = +3, and the x-axis.

36/5