How do you find the area between two curves using integrals?

Answer 1

First, you will take the integrals of both curves. Next, you will solve the integrals like you normally would. Finally, you will take the integral from the curve higher on the graph and subtract the integral from the lower integral.

Take for example we have two functions

#f(x)=32-x^2# and #g(x)=x^2#

Without any limits given we assume they want the area between the points that the two functions intersect so we set the two functions equal and solve.

#32-x^2=x^2# solves to #x = 4# and #x=-4# so they become our bounds

Do to #f(x)# is higher on the positive x-axis, we subtract #g(x)# from it.

Now that we know the bounds and the order to subtract, we can setup the integral.

#int_-4^4f(x)-g(x) dx= int_-4^4 32-x^2-x^2 #

And now we solve like a normal integral.
#F(x) = 32x-2/3x^3 +C#
#F(4)-F(-4)=170 2/3#

Here is a graphical example of a more complicated problems:

The more general form of area between curves is:

#A=int_a^b|f(x)-g(x)|dx#
because the area is always defined as a positive result.

So for this problem, you need to find all intersections between the 2 functions (we'll call red #f(x)# and blue #g(x)# and you can see that there are 4 at approximately: #-6.2#, #-3.5#, #-.7#, #1.5#. So, we look at which function is greater on those intervals for the full integral:

#A=int_(-6.2)^(-3.5)[f(x)-g(x)]dx+int_(-3.5)^(-.7)[g(x)-f(x)]dx+int_(-.7)^(1.5)[f(x)-g(x)]dx#

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Answer 2

To find the area between two curves using integrals, follow these steps:

  1. Identify the Intersection Points: Determine the x-values where the two curves intersect by setting them equal to each other and solving for ( x ).

  2. Identify the Boundaries: Determine the x-values for the interval over which you want to find the area. This interval is typically between the intersection points found in step 1.

  3. Set Up the Integral: Decide which curve is the upper curve ( y = f(x) ) and which is the lower curve ( y = g(x) ) within the interval. The area ( A ) between the two curves from ( x = a ) to ( x = b ) is given by: [ A = \int_{a}^{b} [f(x) - g(x)] , dx ] If ( f(x) ) is always above ( g(x) ) on the interval, then the integral simplifies to: [ A = \int_{a}^{b} [f(x) - g(x)] , dx ]

  4. Evaluate the Integral: Integrate the expression from step 3 over the interval determined in step 2 to find the area.

    [ A = \int_{a}^{b} [f(x) - g(x)] , dx ]

    Compute this integral to get the area between the two curves.

By following these steps, you can find the area between two curves using integrals.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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