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AP Calculus AB: 10.2.1 The Area between Two Curves

Mathematics7 CardsCreated 3 months ago

This section explains how to calculate the area between two curves by integrating the difference between the upper and lower functions over a given interval. It emphasizes correct setup of the integral, using limits of integration from x-values where the region begins and ends, and applying the Fundamental Theorem of Calculus for evaluation.

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The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area of a region bounded by the graphs of two functions, take the definite integral of the difference of the two functions on the appropriate interval.

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Key Terms

Term
Definition

The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area...

note

  • Notice that the area between these two regions is equal to the area underneath the top curve minus the area underneath the bottom curve. Si...

What is the area between the graphs of the functions f (x) = 5 sin x and g (x) = e ^x − 3, between x = 1 and x = 2?

2∫1 (5sinx−e^x−3)dx

What is the area between the curves f(x)=2−x and g(x)=x^2 between x=−2 and x=1?

A=4 1/2

What is the area between the functions g(x)=cos2x and h(x)=5x, between x=3 and x=4?s

.804

Which of the following integrals represents the area between the graphs of the functions f (x) = |x| and g (x) = x^ 2 + 5, between x = −1 and x = 2?

2∫−1 [x^2+5−|x|]dx

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AP Calculus AB: 10.2.1 The Area between Two Curves
TermDefinition

The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area of a region bounded by the graphs of two functions, take the definite integral of the difference of the two functions on the appropriate interval.

note

  • Notice that the area between these two regions is equal to the area underneath the top curve minus the area underneath the bottom curve. Since the sum of two integrals is equal to the integral of the sum, you can combine the two integrals into one.

  • Once you set up the integral, you can use the
    fundamental theorem of calculus to evaluate it.

  • When setting up the integral, remember to take the equation for the upper curve and subtract the equation for the lower curve.

  • The limits of integration are the x-values where the region begins and ends.

  • Notice that the integral itself is very basic. Once simplified, it is just the integral of a polynomial equation. You can evaluate the integral piece by piece.

What is the area between the graphs of the functions f (x) = 5 sin x and g (x) = e ^x − 3, between x = 1 and x = 2?

2∫1 (5sinx−e^x−3)dx

What is the area between the curves f(x)=2−x and g(x)=x^2 between x=−2 and x=1?

A=4 1/2

What is the area between the functions g(x)=cos2x and h(x)=5x, between x=3 and x=4?s

.804

Which of the following integrals represents the area between the graphs of the functions f (x) = |x| and g (x) = x^ 2 + 5, between x = −1 and x = 2?

2∫−1 [x^2+5−|x|]dx

What is the area between the curves f (x) = 3 and g (x) = x ^2 + 4 between x = 0 and x = 3?

A = 12