Area Between Curves Calculator
Find the area between two curves with step-by-step setup, top-minus-bottom logic, intersection checks, shaded graphs, and student-friendly explanations.
Background
The area between two curves is found by integrating the vertical distance between them. In most calculus problems, that means identifying the upper curve, subtracting the lower curve, and integrating over the correct interval.
How to use this Area Between Curves Calculator
- Enter two functions using x as the variable.
- Enter the left and right bounds of the interval.
- Click Calculate & Shade Area to compute the approximate area.
- Use the graph to see which region is being measured.
- Check the warning section to see whether the curves cross inside the interval.
How this calculator works
- The calculator evaluates both curves over the selected interval.
- It compares the curves to decide which one is above the other across the interval.
- It sets up area as the integral of the vertical distance between the curves.
- If the curves cross, it warns that the interval may need to be split.
- It approximates the area using Simpson’s rule or the trapezoidal rule.
- It draws a shaded graph so students can connect the calculation to the region.
Formula & Concepts Used
Area between curves: A = ∫[top curve − bottom curve] dx
With functions: A = ∫ₐᵇ |f(x) − g(x)| dx
If one curve stays above the other: A = ∫ₐᵇ (f(x) − g(x)) dx
If curves cross: split the interval at intersection points and add the positive areas.
Numerical approximation: Simpson’s rule or trapezoidal rule estimates the definite integral.
Example Problems & Example Problems & Step-by-Step Solutions
Example 1: Parabola above the x-axis
Find the area between:
f(x)=4-x², g(x)=0, from x=-2 to x=2
Since 4-x² is above the x-axis on this interval:
A = ∫₋₂² (4-x²) dx
A = [4x - x³/3]₋₂² = 32/3 ≈ 10.667
Example 2: Line above a parabola
Find the area between:
f(x)=x+2, g(x)=x², from x=0 to x=2
On this interval, x+2 is above x².
A = ∫₀² [(x+2)-x²] dx
A = [x²/2 + 2x - x³/3]₀² = 10/3 ≈ 3.333
Example 3: Curves that cross
Find the area between:
f(x)=x, g(x)=x², from x=-1 to x=2
The curves intersect at x=0 and x=1, so split the interval.
A = ∫₋₁⁰ (x²-x)dx + ∫₀¹ (x-x²)dx + ∫₁² (x²-x)dx
A = 5/6 + 1/6 + 5/6 = 11/6 ≈ 1.833
FAQs
What does area between curves mean?
It means the amount of space trapped between two graphs over a given interval.
Why do we subtract bottom from top?
At each x-value, the vertical distance between the curves is the upper y-value minus the lower y-value.
What if the curves cross?
If the curves cross, the top and bottom curves may switch. In that case, split the interval at the intersection point.
Does this calculator give exact answers?
This v1 calculator focuses on numerical area approximations and visual understanding.
Exact symbolic antiderivatives can be added later.
Can I use trigonometric functions?
Yes. Common functions such as sin(x),
cos(x), and simple powers are supported.
Example 1: Parabola above the x-axis
Find the area between:
f(x)=4-x², g(x)=0, from x=-2 to x=2Since 4-x² is above the x-axis on this interval:
A = ∫₋₂² (4-x²) dx
A = [4x - x³/3]₋₂² = 32/3 ≈ 10.667
Example 2: Line above a parabola
Find the area between:
f(x)=x+2, g(x)=x², from x=0 to x=2On this interval, x+2 is above x².
A = ∫₀² [(x+2)-x²] dx
A = [x²/2 + 2x - x³/3]₀² = 10/3 ≈ 3.333
Example 3: Curves that cross
Find the area between:
f(x)=x, g(x)=x², from x=-1 to x=2The curves intersect at x=0 and x=1, so split the interval.
A = ∫₋₁⁰ (x²-x)dx + ∫₀¹ (x-x²)dx + ∫₁² (x²-x)dx
A = 5/6 + 1/6 + 5/6 = 11/6 ≈ 1.833
What does area between curves mean?
It means the amount of space trapped between two graphs over a given interval.
Why do we subtract bottom from top?
At each x-value, the vertical distance between the curves is the upper y-value minus the lower y-value.
What if the curves cross?
If the curves cross, the top and bottom curves may switch. In that case, split the interval at the intersection point.
Does this calculator give exact answers?
This v1 calculator focuses on numerical area approximations and visual understanding. Exact symbolic antiderivatives can be added later.
Can I use trigonometric functions?
Yes. Common functions such as sin(x), cos(x), and simple powers are supported.
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