Calculating the area between curves is a key skill in calculus. It involves integrating the difference between two functions over a specific interval. This technique allows us to find the size of regions bounded by multiple curves.
Choosing the right variable to integrate with respect to is crucial. We need to consider the orientation of the curves and pick the approach that leads to simpler expressions. For complex regions, we might need to break them into smaller, more manageable sub-areas.
Area between Curves
Area between two curves
Top images from around the web for Area between two curves HartleyMath - Area Between Curves View original
Is this image relevant?
AreaBetweenCurvesIntegral | Wolfram Function Repository View original
Is this image relevant?
calculus - Area by integration of the finite region bound by the two curves. - Mathematics Stack ... View original
Is this image relevant?
HartleyMath - Area Between Curves View original
Is this image relevant?
AreaBetweenCurvesIntegral | Wolfram Function Repository View original
Is this image relevant?
1 of 3
Top images from around the web for Area between two curves HartleyMath - Area Between Curves View original
Is this image relevant?
AreaBetweenCurvesIntegral | Wolfram Function Repository View original
Is this image relevant?
calculus - Area by integration of the finite region bound by the two curves. - Mathematics Stack ... View original
Is this image relevant?
HartleyMath - Area Between Curves View original
Is this image relevant?
AreaBetweenCurvesIntegral | Wolfram Function Repository View original
Is this image relevant?
1 of 3
Find area between two curves by integrating difference of upper and lower functions
Integrating with respect to x: A = ∫ a b [ f ( x ) − g ( x ) ] d x A = \int_{a}^{b} [f(x) - g(x)] dx A = ∫ a b [ f ( x ) − g ( x )] d x
f ( x ) f(x) f ( x ) represents upper function, g ( x ) g(x) g ( x ) represents lower function
a a a and b b b are x-coordinates of intersection points (endpoints of interval)
Integrating with respect to y: A = ∫ c d [ h ( y ) − j ( y ) ] d y A = \int_{c}^{d} [h(y) - j(y)] dy A = ∫ c d [ h ( y ) − j ( y )] d y
h ( y ) h(y) h ( y ) represents right function, j ( y ) j(y) j ( y ) represents left function
c c c and d d d are y-coordinates of intersection points (endpoints of interval)
Determine intersection points of curves to establish limits of integration
Equate functions and solve for variable of integration to find intersection points (x x x or y y y values)
Select appropriate variable of integration based on orientation of curves
Choose x x x if curves are primarily vertical (functions of x x x )
Choose y y y if curves are primarily horizontal (functions of y y y )
The area between curves forms a bounded region
Compound regions with intersecting curves
Divide compound region into smaller, simpler sub-regions
Find all intersection points of curves involved
Use vertical or horizontal lines at intersection points to create sub-regions
Evaluate area of each sub-region individually
Identify upper and lower functions (or right and left) for each sub-region
Integrate difference of functions for each sub-region
Add areas of all sub-regions to calculate total area of compound region
A t o t a l = A 1 + A 2 + . . . + A n A_{total} = A_{1} + A_{2} + ... + A_{n} A t o t a l = A 1 + A 2 + ... + A n
A 1 , A 2 , . . . , A n A_{1}, A_{2}, ..., A_{n} A 1 , A 2 , ... , A n represent areas of individual sub-regions
Sum of sub-region areas equals total area of compound region
Consider using integration by substitution for more complex functions
Variable selection for area calculations
Assess orientation of curves to choose appropriate variable of integration
If curves are easily expressed as functions of x x x , integrate with respect to x x x
If curves are easily expressed as functions of y y y , integrate with respect to y y y
Circumvent integrating with respect to a variable that necessitates solving for the other variable in terms of it
Leads to intricate expressions and convoluted integration
Opt for the variable that yields the most straightforward expressions for upper and lower functions (or right and left)
Uncomplicated expressions simplify integration and minimize sub-regions
Fewer sub-regions reduce complexity of area calculation
Advanced techniques for area calculations
Use cross-sectional area method for solids of revolution
Apply double integration for more complex regions
Utilize polar coordinates for circular or radial symmetry